Reconstruction and stability in Gel'fand's inverse interior spectral problem
Roberta Bosi Yaroslav Kurylev Matti Lassas












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                            Roberta Bosi, Yaroslav Kurylev, and Matti Lassas
                            December 10, 2019
                        =============================================================================
                        
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                                Assume that M is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter,  Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian Δ_g on M as well as the corresponding eigenfunctions restricted on an open set in M. We then construct a stable  approximation to the manifold (M,g). Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from M  when the above data are given with a small error. We give an explicit loglog-type  stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gel'fand's inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric Boundary Control method.
                                
                                
                                § INTRODUCTION
                                
                                
                                 §.§ Inverse interior spectral data and classes of manifolds
                                 Let (M,g,p) be a pointed   compact Riemannian manifold,
                                that is, (M,g) is a compact Riemannian manifold without boundary and p∈ M  is a point on M.
                                Let Δ _g be the Laplace operator on (,g), 
                                with 0=_0 <_1≤_2≤… being its eigenvalues and
                                φ_j, j=0,1,2,… being the complete sequence of
                                L^2(M)-orthonormal
                                eigenfunctions satisfying -Δ_g φ_j=λ_j φ_j on M.
                                
                                 Let (M, g, p) be an n dimensional compact pointed
                                manifold with n≥ 2. Let r_0>0. Then
                                
                                (i) The pair, consisting of the ball  (B(p, r_0),g|_B(p,r_0)) on the  Riemannian manifold M
                                and the sequence
                                  {(_j, φ _j|_B(p,r_0)); j=0, 1,2,…}  of eigenvalues and eigenfunctions,
                                   is called the interior spectral data (ISD) of (, g, p).
                                
                                (ii)
                                The pair, consisting of the ball  (B(p, r_0),g|_B(p, r_0)) and a finite collection  {(_j, φ _j|_B(p, r_0)),j=0, 1,2,…,J} of the J+1  first eigenvalues and eigenfunctions,
                                 is called the finite interior spectral data (FISD) of (, g, p).
                                 The interior Gel'fand inverse spectral problem is that of the reconstruction of (M, g) from
                                its ISD. It was solved in <cit.>, <cit.>. In this paper we consider the problem 
                                of an approximate reconstruction of (,g) when we
                                know only its FISD,
                                namely,  the first eigenvalues,
                                _j < δ^-1
                                  with some small δ∈ (0,1)
                                and the corresponding eigenfunctions of φ_j|_B(p, r_0).
                                 Furthermore, we assume that we know all these objects with some error. However, due to the well-known ill-posedness of inverse problems, to achieve this 
                                goal one needs to assume that the manifold to be approximately reconstructed should lie in a properly 
                                chosen class of manifolds.  In this paper we concentrate on an appropriate Gromov's class of 
                                pointed manifolds.
                                
                                
                                Next we define a class of manifolds satisfying geometric bounds, in terms of the constants 
                                R, D, i_0, and n, and the radius r_0. Those constants have to be consider as global parameters in all calculations.
                                
                                
                                
                                
                                
                                (Riemannian manifolds of bounded geometry). For any n∈_+ and R>0,D>0, i_0>0,
                                _ n:=_ n(R, D, i_0 ) consists of n-dimensional pointed 
                                compact Riemannian manifolds (,g,p)
                                such that
                                       i) ∑_j=0^3 ∇^j Ric(,g)_L^∞(M,g)≤ R, 
                                
                                       ii)  (,g) ≤ D, 
                                       iii)   (,g) ≥ i_0.
                                
                                Here Ric(, g)=Ric^M_jk stands for the Ricci curvature of M, 
                                (M, g) for the diameter of M,
                                and (,g)  for the injectivity radius of (M,g). At last,
                                 ∇ stands for the covariant derivative on (M,g).  
                                
                                The norm of ∇^j Ric(,g)
                                
                                is computed using the metric g, e.g. ∇Ric^M=(g^ii'g^jj'g^kk'(∇_i Ric^M_jk)(∇_i'Ric^M_j'k'))^1/2. We recall that 
                                a pointed 
                                compact Riemannian manifold (,g,p) consists of a manifold M, its Riemannian metric g,
                                and an arbitrary point p∈ M. This definition is used as we specify the point p near 
                                which the values of the eigenfunctions are measured.
                                
                                
                                In the future, without loss of generality, we assume 
                                
                                
                                r_0 < min(i_0/2, π/2 √(K),1).
                                
                                 Here K is the bound for
                                the sectional curvature on _ n.  The bound K depends only on R, D, i_0, and n,
                                see (<ref>). This makes it possible to use
                                in B(p, r_0) the Riemannian normal coordinates which allows us to compare interior spectral data
                                of different manifolds in _ n.
                                To formalise the above, let B(r_0) ⊂^n be an Euclidian ball of radius r_0
                                  and h be 
                                some Riemannian  coordinates in B(r_0) making it a ball of radius r_0 with respect 
                                to h. Let
                                 be a collection of elements  (Data Sequences)
                                DS=( (B(r_0),h) , {(μ_j, ψ_j|_B(r_0))}_j=0^∞ )
                                
                                where 0=μ_0 < μ_1 ≤μ_2 ≤…,  μ_j →∞, and
                                ψ_j∈  L^2(B(r_0), h). 
                                
                                
                                
                                
                                
                                
                                 (Interior spectral topology.)
                                Let δ>0. For i=1,2, consider the collections DS^i  ∈.
                                
                                We say that DS^1 and DS^2
                                are δ-close
                                if the following is valid:
                                
                                There are  P∈_+  and disjoint intervals
                                
                                I_p = (a_p,b_p) ⊂ ( -δ, δ^-1+δ),
                                   p=0, 1,…, P,
                                
                                such that
                                
                                
                                i) b_p-a_p <δ.
                                ii) For any μ_j^i,  i=1,2 with  |μ_j^i| < δ^-1
                                there is p such that
                                μ_j^i ∈  I_p.
                                
                                
                                
                                iii)  For p=0,   n_0^i=1. For any p ≥ 1, the total number n_p^i of elements in sets 𝒥^i_p={j∈_+; μ_j^i∈ I_p}
                                coincide, i.e.
                                n_p^1 = n_p^2), and satisfies n_p^1 = n_p^2≥ 1.
                                iv)    There is an orthogonal matrix O ∈ O(n), 
                                
                                such that the  metrics O_* h_1 and h_2  are Lipschitz δ-close
                                on B(r_0), i.e., for
                                any x∈ B(r_0) andξ=(ξ^1, …, ξ^n) ∈^n, ξ≠ 0,  we have
                                (1+δ)^-1≤(O_* h_1)_jk(x) ξ^j ξ^k/(h_2)_jk(x) ξ^j ξ^k≤ 1+δ,
                                
                                v) For any p there is a unitary matrix
                                
                                    A_p =[a^(p)_jk]_j,k∈𝒥_p∈ U(n_p),
                                
                                such that
                                      A_p   (O_*  Ψ_p^1)- Ψ_p^2_(L^2(B(r_0),h_2))^n_p≤δ,
                                      A_p^-1  ( (O^-1)_* Ψ_p^2)- Ψ_p^1_(L^2(B(r_0),h_1 ))^n_p≤δ.
                                
                                Here,Ψ_p^i is the vector-function
                                {ψ_j}_j ∈𝒥^i_p.
                                 Note that above the number P indicates 
                                
                                
                                how many groups of eigenvalues are clustered to satisfy conditions i-iv. Moreover, for two sequences DS^1 and DS^2, the above conditions i-iv may be valid
                                with several  different values of P and intervals  I_p,p=1,2,…, P. Condition v) can be interpreted as the closedness of the Riesz projectors corresponding to Δ_g_i onto I_p.
                                
                                
                                 We note that in a more restricted context of Gelfand's inverse problem for a Schrödinger operator with simple spectrum in a domain in ^n
                                 a similar topology was introduced  in
                                <cit.>.
                                
                                
                                
                                
                                 §.§ The main results
                                
                                
                                To formulate our result on an approximate reconstruction, we use
                                the  Gromov-Hausdorff distance.
                                
                                 (GH-topology, see e.g. <cit.>, <cit.>).
                                  Let (X^i, d^i, p^i), i=1,2 be pointed
                                compact metric spaces. Then the  pointed Gromov-Hausdorff
                                distance d_GH(X^1, X^2) is the infimum of
                                all >0  such that there is a metric space
                                (Z,d_Z) and isometric embeddings
                                i_1:X^1→ Z and i_2:X^2→ Z which satisfy
                                
                                d_H( i_1(X^1),  i_2(X^2))<,   d_Z( i_1(p^1),  i_2(p^2))<.
                                
                                Here d_H denotes the Hausdorff distance in Z, see <cit.>.
                                
                                
                                The   main result of the paper is:
                                
                                 Let n≥ 2, R, D, i_0 and r_0 satisfying (<ref>)
                                be given. Then there  exist>1 and >0,
                                depending only on n, R, D, i_0 and r_0, such that the following is true:
                                
                                
                                Let (M^(1),g^(1),
                                p^(1)),(M^(2),g^(2),
                                p^(2)) ∈ M_n.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                Assume that  the
                                interior spectral data
                                of the operators -Δ_g^(i)  on M^(i) in the balls B^(i)=B_M^(i)(p^(i),r_0)⊂ M^(i), that is, the collections
                                (  (B^(i), g^(i))  , {(λ_j^(i), φ^(i)_j|_B^(i)) ;  
                                j=0,1,2,…})
                                
                                are δ-close, in the sense of Definition <ref>,
                                with
                                0<δ≤exp(-e).  Then
                                
                                d_GH((M^(1),p^(1)),(M^(2),p^(2)))≤(ln(ln1/δ))^-.
                                 The above stability estimate is log-log  type. It is not known if this type of result is optimal, but the counterexamples of Mandache <cit.> for equivalent inverse problem show that the stability result can not be better than logarithmic.
                                
                                
                                The proof of Theorem <ref> is constructive, and 
                                is based on the following 
                                
                                 result on the reconstruction of the manifold from the data.
                                 Below, when we state that a manifold (M^*,g^*) can be constructed from the data, 
                                we mean 
                                that there is a sequence of steps, where we solve a finite number of  
                                quadratic minimization problems in  
                                
                                finite dimensional spaces, choose elements from finite sets or compute certain explicit functions. 
                                Indeed, we do the following steps. First, we solve quadratic minimization problems in finite dimensional vector spaces 
                                (that are equivalent to solving linear equations) to find the finite sequences 
                                (d^a_j(,i))_j=0^∈^, where (,i)  run over a finite index set, see Theorem <ref>. Second, we use these sequences to compute approximative volumes 
                                ^a(M^*_ (i)(β)), of subsets of M, where (i,β) runs over a finite index set, see Lemma <ref>.
                                Third, we choose the set of admissible indexes β for which the approximative volumes are larger than a certain threshold value, see Definition <ref>. The admissible indexes are used in Section <ref>
                                and Lemma <ref> to define a finite set 
                                of piecewise constant functions, R^*_M, that  approximate the collection of the interior distance functions.
                                
                                
                                
                                Using the finite set R^*_M  and a modified version of the construction given in <cit.> we construct 
                                
                                a finite metric space (M^*,d^*), that approximates the Riemannian manifold (M,_g)  in Gromov-Hausdorff sense. Let n≥ 2, R, D, i_0 and r_0 satisfying (<ref>)
                                be given. 
                                Then there exists a constant
                                δ^* = δ^*(n, R, D, i_0, r_0)  and  positive constants <1 and >1, depending  only onn, R, D, i_0 and r_0,
                                such that,
                                for all δ with
                                
                                0<δ≤δ^*,
                                
                                the following is true:
                                
                                
                                Assume that (M,g,
                                p) ∈ M_ n and we are given a collection
                                (  (B(r_0), g^a)  , {(μ_j, φ^a_j|_B(r_0)) ;  
                                j=0,1,2,…,J})
                                
                                that  is δ-close, in the sense of Definition <ref>,
                                 to interior spectral data
                                of the operator -Δ_g  on M.  Using the data (<ref>) we can construct  a pointed metric space  (M^*, d^*,p^*)
                                such that
                                  
                                d_GH(M, M^*)≤,   where  =(ln(ln1/δ))^- .
                                 We note that in Proposition <ref> that value of J  is not fixed,
                                but it just has to be so large that every j, for which the eigenvalue λ_j  satisfies
                                λ_j<δ^-1+δ, fulfils the inequality j≤ J, see (<ref>).
                                
                                The relation J,  i.e., the number of eigenvalues, and the accuracy parameter  δ
                                is discussed in Remark <ref> below. 
                                
                                
                                
                                
                                
                                 A note on the used constants. In the main part of the  paper, we will make frequent use of constants 
                                c, C, C_1, C_2, etc. 
                                
                                These constants will depend only on the geometric bounds n, R, D, i_0, r_0, see Definition <ref>, but may change in their value from line to line. The constants that
                                depend only on the geometric bounds  n, R, D, i_0, r_0 will be called `uniform constants'. 
                                
                                
                                
                                
                                When we define a constant for the first time, we specify whether it is uniform or not and write its further dependencies in parenthesis. For example the constant C(s,m) (or C_s,m)  depends also on
                                s and m.  Before Appendix we have collected a  table on the locations where the  constants C_k and c_k are defined. Conventions for constants in the Appendix are explained  in each subsection.
                                
                                Also M_ n
                                is the closure of M_ n in the GH topology.
                                Parameter K above and in Corollary 
                                <ref> below is the bound for the sectional
                                curvature which is uniform, see (<ref>), on M_ n.
                                
                                 As shown in section 2.1
                                the class M_ n is compact.
                                Thus, when checking  condition v) of definition <ref> it is sufficient
                                to use the standard L^2- norm on B(r_0).  
                                 Recall that the for  pointedC^1-diffeomorphic manifolds (M_1,p_1) and (M_2,p_2)  the Lipschitz distance 
                                is
                                
                                d_L((M_1,p_1),(M_2,p_2))      =inf_F:M_1→ M_2( ln(Lip(F))+ln(Lip(F^-1))+
                                       +d_M_2(p_2,F(p_1))+d_M_1(p_1,F^-1(p_2)))
                                
                                where the infimum is taken over  bi-Lipschitz maps F:M_1→ M_2 and   Lip(F) is the Lipschitz-constant of the map F, see <cit.>.
                                Inequality (<ref>) combined with the
                                sectional curvature bound (<ref>) and
                                the solution of the geometric Whitney problem <cit.> implies the following stable construction result for the manifold M in the Lipschitz topology. 
                                
                                Let (M,g,p)∈ M_ n, δ>0, and the metric space M^*
                                be as in 
                                 Proposition <ref>. 
                                
                                
                                Using M^* one can  construct  
                                a smooth pointed Riemannian manifold (N,g_N,p_N) such that
                                |Sec(N)|≤ K,   inj(N)≥min{ ( K)^-1/2, (1- K^1/3σ_0^2/3) i_0},
                                and M and N are diffeomorphic. Moreover,
                                
                                d_L( (M,p),(N,p_N))≤ K^1/3σ_0^2/3,   σ_0=(ln(ln1/δ))^-,
                                
                                Here Sec stands for the sectional curvature and , are uniform constants.
                                 Instead of eigenvalues and eigenfunctions 
                                one can deal with the heat kernels
                                H_M(x, y, t) of ∂_t-Δ_g, cf. <cit.>, <cit.>, <cit.>.
                                Definition <ref> can be reformulated e.g. as
                                H_M^(1)-H_M^(2)_C(B(p, r_0)^2 × (δ, ∞)) < δ.
                                An analog of  Theorem <ref> can be obtained. However, we do  not dwell on this
                                issue in the  paper.
                                
                                
                                To complete this section we recall that stability in the corresponding direct spectral
                                problem is well-known, see e.g. <cit.>.
                                 In particular, let M be a compact manifold equipped with metrics g_ℓ, ℓ=1, 2, …, and g_0. Let  a,b∉σ(-Δ_g_0). Denote by P_ℓ,  P_0 the spectral 
                                 orthoprojectors in 
                                 L^2(M, g_ℓ), L^2(M, g_0) on the interval [a, b].
                                  Then it follows from Theorems  IV.3.16 and VI.5.12 of <cit.> that if
                                g_ℓ-g_0_L^∞(M)→ 0 as ℓ→∞, 
                                then P_ℓ-P_0_L^2(M,g_0)→ L^2(M,g_0)→ 0. This implies
                                that the ISD of (M,g_ℓ) converges to the ISD of (M,g_0).
                                
                                
                                
                                 
                                
                                 §.§ Earlier results and outline of the paper
                                
                                
                                The Gel'fand inverse problem, formulated by I. M. Gel'fand in 50s <cit.>, is the problem of determining 
                                the coefficients of a second order
                                elliptic differential operator in a domain Ω⊂^n from the boundary spectral data, that is,
                                the eigenvalues and the boundary values of the eigenfunction of the operator.
                                In the geometric Gel'fand inverse problem, a Riemannian manifold with boundary and a metric tensor on it
                                need  to be constructed from similar data.
                                 For Neumann boundary value problem for the operator -Δ_g on manifold M,
                                the boundary spectral data consists of the boundary
                                 M, the eigenvalues λ_j and the boundary values of the  eigenfunction,
                                 φ_j|_ M, j=1,2,… 
                                 The uniqueness of the solution of the Gel'fand inverse problem has been considered in
                                <cit.>.
                                
                                
                                
                                
                                
                                
                                To consider the formulation of the stability of the inverse problems, 
                                let us consider first the Gel'fand inverse on 
                                 a bounded domain  Ω⊂^2 with smooth boundary Ω and a conformally Euclidian metric 
                                 g_jk(x)=ρ(x)^-2δ_jk. Here, ρ(x)>0 is a smooth real valued function. Then the problem has the form
                                      
                                -∑_k=1^2ρ(x)(/ x^k)^2φ_j(x)-λ_jφ_j(x)=0,  in Ω,
                                  ∂_νφ_j|_Ω=0.
                                
                                The problem of determining ρ(x) from the boundary spectral data is ill-posed  in sense of Hadamard:
                                The map from the boundary data
                                to the coefficient ρ(x) is not continuous so that small change in the data can lead to huge errors in the 
                                reconstructed function ρ(x).
                                One way out of this fundamental difficulty 
                                is to assume a priori higher regularity of coefficients, that is
                                a widely used trend in inverse problems for isotropic equations,
                                like (<ref>). This type of results is called 
                                 conditional
                                stability results (see e.g.
                                <cit.>).
                                
                                For inverse problems for general
                                metric  this approach bears significant difficulties.
                                The reason is that the usual C^k norm bounds of coefficients
                                are not invariant and thus this condition does not suit
                                 the invariance of the problem  with respect to
                                 diffeomorphisms. Moreover, if 
                                the structure of the manifold M
                                is not known a priori, the traditional approach can not be used.
                                The way to overcome these difficulties
                                is to impose a priori 
                                 constraints in an invariant form and consider
                                a class of manifolds that satisfy invariant a priori bounds,  for instance
                                for  curvature, second fundamental form, radii
                                of injectivity, etc. Under such kind of conditions,  invariant stability results for various inverse problems  
                                
                                have been proven 
                                
                                in <cit.>.
                                In particular, for the Gel'fand inverse problem for manifolds with non-trivial topology, an abstract, i.e., a non-quantitative 
                                stability 
                                result  
                                was proven in <cit.>. There, it was shown that the convergence of the boundary spectral data implies
                                the convergence of the manifolds
                                with respect to the Gromov-Hausdorff convergence. However, this result was based on compactness arguments
                                and it did not provide any
                                 estimate . In this paper our aim is 
                                
                                to improve 
                                this result and to give explicit estimates for an analogous inverse problem.
                                
                                
                                In this paper we consider a Gel'fand inverse problem for manifolds without boundary. Then, as explained above, instead of assuming that the boundary and the boundary values of 
                                the eigenfunctions are known we assume that we are given a small open ball B⊂ M and
                                the eigenfunctions φ_j are known on this set.
                                Similar type of formulation of the problem with measurements on open
                                sets have been considered in <cit.>.
                                We show that the Interior Spectral Data (ISD), that is, an open set B⊂ M, the eigenvalues λ_j  and the restrictions of the eigenfunctions  φ_j|_B 
                                determine the whole manifold (M,g) in stable way. Also, we quantify this stability by giving explicit
                                inequalities under a priori assumptions on the geometry of M.
                                 We emphasise that we assume that the eigenfunctions are known only on an open subset B of M that may
                                 be chosen 
                                
                                
                                to be arbitrarily small but still e.g. the topology of M is determined in a stable way. We note that this paper is a slightly extended and polished version of our preprint in Arxiv, published on Feb. 25, 2017. We note that in spectral geometry one has studied similar stability problems where the heat kernel are known on the whole
                                 manifold, <cit.>. This data is equivalent to knowing the eigenvalues and the eigenfunctions and the eigenfunctions on the whole manifold.
                                
                                
                                Outline of the paper: Ch. 2 introduces the geometric set-up.
                                Ch. 3 formulates the stability of the unique continuation for the solution of the wave equation together with Corollary <ref>  for its spatial projection v.
                                Ch. 4 presents Thorems <ref> and <ref> proving the construction of the approximate Fourier coefficients of χ_Ωv in the case of respectively exact and approximate FISD. 
                                Ch. 5 shows the related approximate interior distance functions. Ch. 6 collects all the previous inequalities to prove Theorem <ref>
                                and Proposition <ref>.
                                
                                
                                
                                
                                
                                § GEOMETRIC PRELIMINARIES
                                
                                
                                 §.§ Properties of the manifolds of bounded geometry 
                                
                                
                                
                                Here we list some results on 
                                 the class
                                _ n(R, D, i_0),
                                
                                These results can be found in
                                or immediately follow from
                                <cit.> with further improvements in <cit.>.
                                
                                
                                Namely, the class _ n is precompact in GH-topology. Its closure,
                                _ n consists of pointed Riemannian manifolds (, g,p)
                                with g ∈ C^5_*(M) which satisfy (<ref>). Here and later ^* indicates the Zygmund space.
                                
                                
                                We define the norm of the space C^k(M) invariantly by
                                f_C^k(M):=∑_j=0^k  max_x∈ M∇^j f(x)_g,
                                
                                where  the norm is computed using the metric g. Next, for k ∈_+,  β∈ (0, 1], we use the the Zygmund spaces
                                    C^k+β_*(M)=[C^k_1(M),C^k_2(M)]_θ,    
                                    k+β=θ k_1+(1-θ) k_2∈_+,   θ∈ (0,1).
                                
                                Here [·, ·]_θ stands for the interpolation, see e.g <cit.>.
                                
                                Note that, for β∈ (0, 1), the Hölder spaces 
                                 fulfillC^k,β(M)=C^k+β_*(M).
                                
                                
                                To achieve the C^k_*-smoothness of g, one needs some special coordinates, e.g. harmonic coordinates. For any 
                                number Q>1, that we below choose to be Q=2, there is a constant r^(har) depending only on 
                                 n, R, D, i_0, r_0 and Q, such that, for any
                                (M,g,p)∈_n,  q ∈ M, there are Q-harmonic coordinates in B(q,  r^(har)), that we denote by Y:B(q,  r^(har))→^n, U_q=Y(B(q,  r^(har)))
                                that we denote by y.
                                 For Q=2, in these coordinates  the metric tensor g^(har)_jk(x)=(Y^-1)^g  satisfies
                                       2^-1I ≤ (g^(har)_jk(y))_j,k=1^n ≤ 2 I,    for y∈ U_q=Y(B(q,  r^(har)))
                                      g^(har)_jk_C^5_*(U_q)≤ C^(har),
                                with some uniform constant C^(har), see <cit.> and <cit.>.
                                We note that the existence of the harmonic radius r^(har)
                                and the constant C^(har) for which (<ref>) holds for all (M,g,p)∈_n,q ∈ M,
                                is based on compactness results, and therefore the dependency of r^(har)
                                and C^(har) on n, R, D, i_0, r_0 is not explicit.
                                
                                Sometimes, with a slight abuse of notation we identify y∈ U_q with the corresponding point in Y(y)∈ B(q,  r^(har)).
                                
                                The inequality  (<ref>) immediately implies that  the sectional curvature Sec
                                and the Riemannian curvature tensor R_M satisfies
                                
                                 |Sec(M)|≤ K,  R_M≤ K,  ∇ R_M≤ K,
                                
                                whereK  is a uniform constant.
                                
                                
                                For the sake of simplicity, we will work with Hölder rather then Zygmund
                                spaces.
                                It follows from
                                <cit.>, with the terminology described in <cit.>,  that
                                 when  
                                (M_k,g_k,p_k) → (M,g,p) in the GH topology on _ n,
                                then, for all β∈ (0,1), there are C^5,β-smooth diffeomorphism F_k:M_k→ M such that
                                F_*(g_k) → g   in C^4, β(M), as k→∞.
                                
                                Thus,  for any >0, β<1, there is
                                σ=σ(, β) such that we have the following: For all M_1,M_2∈_ n
                                such that d_GH(M^1, M^2) <σ, there is a
                                diffeomorphism F: M^1 → M^2 and
                                g^h_1-F_*(g^h_2)_C^4, β(M^i) <,   i=1, 2,
                                
                                 cf.
                                 <cit.>.
                                Returning to (<ref>), for large k, M_k and M are diffeomorphic,  so that it
                                is possible to use results from <cit.>, see the end of sec. 1.2. This implies stability of the 
                                direct problem
                                in the GH topology on _ n.
                                
                                
                                 Note that we can solve the ordinary differential equations that define the geodesics  in the harmonic coordinates.
                                Then it follows from (<ref>) that
                                there is a uniform constant >1, such that for  any ball B(x,r)⊂ M,  where (M,g,p) ∈_ n, we have1/ r^n ≤vol(B(x, r)) ≤ r^n,    0 ≤ r ≤ D.
                                  Thus,  the volume of balls having radius i_0/2 is bounded below by a uniform constant v_0. 
                                Furthermore,
                                by <cit.>, the class of Riemannian manifolds (M,g) that satisfy (<ref>) 
                                and conditions (,g) ≤ D  and (,g) ≥ v_0 are pre-compact with respect to 
                                the Lipschitz distance d_L((M_1,p_1),(M_2,p_2)), see (<ref>) and the closure of this class consists of C^∞-smooth manifolds
                                with C^1,α-metric. This implies that there is a uniform constant C^(Lip) such that for all 
                                (M_1,g_1,p_1), (M_2,g_2,p_2)∈_n we have
                                
                                d_L((M_1,p_1),(M_2,p_2))≤ C^(Lip).
                                
                                Moreover, by <cit.>, we have that for any >0 there is 
                                ζ()>0,
                                
                                such that for (M_1,g_1,p_1),(M_2,g_2,p_2)  ∈_n we have
                                if d_GH((M_1,p_1),(M_2,p_2))<ζ() then d_L((M_1,p_1),(M_2,p_2))<.
                                We turn now to the spectral properties on (M,g,p)∈_ n.
                                 By <cit.>, the inequality (<ref>) implies 
                                 that the j-th eigenvalue
                                λ_j(M_i,g_i) of the Laplacian on the manifold (M_i,g_I)
                                satisfies
                                
                                
                                    e^-(n+2)C^(Lip)/2λ_j(M_1,g_1)
                                    ≤λ_j(M_2,g_2)≤ e^(n+2)C^(Lip)/2λ_j(M_1,g_1)
                                
                                for all 
                                (M_1,g_1,p_1), (M_2,g_2,p_2)∈_n.
                                
                                
                                
                                
                                
                                Since
                                
                                 
                                the eigenvalues of the manifold  (M_1,g_1)  satisfy the Weyl's asymptotics 
                                _j(M_1)= c__M_1 j^2/n(1+o(1)) as j→∞, then there exists  a uniform constant >1 such that, 
                                
                                1/ j^2/n≤_j(M) ≤ j^2/n,    j∈_+  for all  (M,g,p)∈_ n(R, D, i_0).
                                 
                                Note  that (<ref>) is valid under a  weaker assumption
                                that Ric(M) is bounded from below, see
                                <cit.>.
                                
                                
                                Assume that the collection of
                                  g^a|_B_e(r_0) and ((λ_j^a,φ_j^a |_B_e( r_0)))_j=0^J 
                                 is δ-close to the FISD
                                  g|_B_e(r_0) and (λ_j,φ_j |_B_e( r_0))_j=0^J of the manifold (M,g,p)∈_ n.
                                Then all intervals I_p= (a_p,b_p), p=0,1,…,P   in (<ref>) satisfy b_j≤δ^-1+δ,
                                and thus the index j of any eigenvalue λ_j that is in some of these intervals satisfies
                                by (<ref>) the inequality ^-1 j^2/n≤δ^-1+δ≤  2δ^-1.
                                On the other hand, if j < ( ^-1δ^-1)^n/2, then
                                λ_j<δ^-1.
                                 Thus, without loss of generality, 
                                we can always   assume that the
                                 value of  J in 
                                 
                                 
                                Proposition <ref> 
                                  satisfies
                                
                                ( ^-1δ^-1)^n/2≤ J≤   (2δ^-1)^n/2.
                                
                                Below we will assume that δ < (3)^-1.
                                Then for j≥ 1 we have λ_j≥^-1 and 
                                λ_j>3δ.
                                Next, assume that λ_j  and λ_k with k>j≥ 1 belong in the same interval I_p= (a_p,b_p)
                                with b_p-a_b<δ.
                                Since λ_j≥^-1 >3δ, we have a_p>2δ so that
                                b_p<2a_p.
                                Then by (<ref>) we have
                                
                                
                                
                                
                                
                                    ^-1 k^2/n≤λ_k≤ 
                                    b_p≤ 2a_p≤ 2λ_j≤ 2 j^2/n,
                                
                                implying
                                
                                j< k≤ (2^1/2)^n j.
                                
                                Next, instead of harmonic coordinates, we can use coordinates made of the eigenfunctions
                                 φ_j. It turns out, cf. <cit.>, that in a neighbourhood of any x ∈ M
                                 there are φ_j(1; x), …, φ_j(n; x) which form C^6_*-smooth coordinates.
                                 Moreover, by the compactness arguments, there are uniform  constants r and C so that 
                                 these coordinates are well defined in any ball B(x, r)⊂ M, where (M,g,p) ∈_ n, and
                                the metric tensor g in these coordinates satisfies (<ref>).
                                There is also a uniform number N ∈_+, such that   we can takej(ℓ; x) ≤ N, ℓ=1, …, n.
                                
                                Next, using  ((_j, φ_j))_j=0^∞,
                                we introduce the Sobolev spaces
                                H^s(M),   s ∈,
                                
                                f(x) =∑_j=0^∞ f_j φ_j(x) ∈ H^s(M)  iff  f^2_H^s:=∑_j=0^∞⟨|_j ^s |f_j|^2 <∞,
                                
                                where ⟨| = (1+^2)^1/2.
                                
                                
                                
                                 §.§ Distance coordinates
                                
                                Recall that there are harmonic coordinates in B(x, r^(har)) ball near any x ∈ M
                                 ∈_ n,  see (<ref>). In the Proposition below we use
                                such coordinates as background coordinates near x.
                                
                                 Below, we say that a subset Y⊂ X is a τ-net in 
                                the metric space X
                                if
                                
                                the  union of the balls B_X(y,τ), y∈ Y, contains the whole space X.
                                Also, we say that Z⊂ X is τ-separated, if for all z_1,z_2∈ Z,
                                z_1≠z_2  we have d_X(z_1,z_2)≥τ. Observe that if Z⊂
                                X  is a maximal τ-separated subset of X (maximal in the sense that any
                                other  τ-separated subset of X that contains Z  has to be equal to Z)
                                 
                                , then it is a τ-net in X.
                                
                                There are uniform  constantsτ_0,ρ_0<min{ r^(har)/4, r_0/128} and uniform constants
                                
                                 L∈_+  and, , >0 depending only on n, R, D, i_0 and r_0,
                                
                                 such that,
                                for any  (M,g,p)∈_ n(R, D, i_0) 
                                
                                the following holds true:
                                
                                There is a τ_0-net in B(p, r_0/4) with at most L-1 points.
                                 Let  {z_1, …, z_L-1}⊂  B(p, r_0/4)  be 
                                 
                                  an arbitrary collection of points that is aτ_0-net in B(p, r_0/4).
                                Then,
                                
                                
                                (i) For all x ∈ M, there are n  points z_j(i)∈ Z, 
                                j(i)=j(i; x), i=1,2,…,n  such that
                                the map X:B(x,ρ_0)→^n,
                                
                                X:y =(y^1, …, y^n)↦ (d_M(y, z_j(1)),d_M(y, z_j(2)), …, d_M(y, z_j(n))),
                                
                                for coordinates where X:B(x,ρ_0)→ X(B(x,ρ_0)) is a Lipschitz-smooth diffeomorphism 
                                
                                and
                                 DX_L^∞(B(x,ρ_0))+ DX^-1_L^∞(X(B(x,ρ_0))))≤.
                                 
                                where  the norms are computed using the metric g on M and the Euclidean norm in ^n. 
                                Moreover, z_j(i) can be chosen so that d(x, z_j(i)) > r_0/ 16 and
                                the metric tensor (g_ij)_i,j=1^n=X_*g in these coordinates satisfies
                                      ^-1 I ≤ (g_ij(z))_i,j=1^n≤ I,   z∈ X(B(x, ρ_0)).
                                
                                
                                
                                
                                (ii) The map H:M→^L-1, defined by H^L(x)=(d_M(x,z_j))_j=1^L-1, satisfies
                                  1/≤|H^L(x)-H^L(y)|/d(x,y)≤,  for all   x,y∈ M, x≠y.
                                  Here as a norm in (<ref>) we can take e.g. the Euclidian norm in
                                ^L-1.
                                 
                                 Proof.
                                 Let us first consider one pointed manifold (M,g,p)∈_ n. 
                                  Let us consider 
                                the extended exponential map
                                 
                                    F:TM→ M× M,   F(x,ξ)= (x,exp_x(ξ)).
                                 
                                Inequalities  (<ref>)  and (<ref>) imply
                                that in the set S={(x,ξ)∈TM: ξ≤ 2D} the map F is C^2-smooth and its norm
                                 in C^2(S) is bounded by a uniform constant. The proof this is analogous to that of Lemma 2 in
                                 <cit.>. 
                                Let
                                  x_0 ∈ M and γ_x_0,ξ_0([0,s_0]), ξ_0∈ S_x_0M  be a shortest geodesic
                                  from x_0 to p where s_0=d(x_0,p). When s_0≥ r_0/2, choose s_1=s_0-  r_0/5,
                                  and when  s_0<  r_0/2,  choose s_1=s_0+  r_0/5. Then the  point p_1=γ_x_0,ξ_0(s_1)
                                  satisfies 
                                  p_1 ∈ B(p,  r_0/5),
                                and d(x_0, p_1) ≥ r_0/5. As r_0<i_0, we see that the geodesic γ_x_0,ξ_0([0,s_1+ r_0/5])
                                is a length minimising  geodesic  between its endpoints. In particular, this implies that 
                                γ_x_0,ξ_0([0,s_1]) 
                                continues behind
                                p_1 as a shortest   curve between  its points. 
                                 
                                
                                As in <cit.>,
                                (see also <cit.> where related results are proven with lower regularity assumptions), we see that (<ref>)  and (<ref>) imply
                                that there is a uniform constant r_*∈ (0,r_0/100) 
                                 such that following is true.  Let 𝔹=B_TM((x_0,s_1ξ_0),r_*) be the ball of radius r_* and center (x_0,s_1ξ_0) defined in the tangent bundle 
                                 (TM,g) using the Sasaki metric. Then for vectors (x,sξ)∈𝔹, where  ξ∈ S_xM, s>0, the geodesics γ_x,ξ([0,s])
                                 are length minimizing curves between their end points. Moreover,
                                 the exponential map in 𝔹, that is,
                                 
                                    F:𝔹→ F(𝔹)
                                
                                 is a diffeomorphism and satisfies dF≤ C in 𝔹, and  (dF)^-1≤ C in  F(𝔹),  and F(𝔹) ⊂ M× B(p_1, r_0/100).  Here,
                                M× B(p_1, r_0/100)
                                 ⊂ M× B(p, r_0/4).
                                 In particular, d(x,z)=|F^-1(x,z)| for (x,z)∈ F(𝔹).
                                
                                
                                
                                 Let ξ_j∈ S_x_0M and t_j>0, j=1,2,…,n be such that 
                                 ξ_j-ξ_0<r_*/s_1,  ξ_j-ξ_k>r_*/(8s_1) for j≠k, and  |t_j-s_1|<r_*/8.
                                 
                                 Then s_1ξ_j∈𝔹.
                                 Let z_j=exp_x_0(t_jξ_j)∈ B(p, r_0/4).
                                 We see that  ∇ d_M( ,z_j)|_x_0=-ξ_j
                                 and  (dF|_x_0)^-1≤ C. 
                                
                                 
                                Inverse function theorem, see e.g. <cit.>, and the facts that (dF|_x_0)^-1≤ C
                                and that F  has a uniformly bounded C^2-norm in S,
                                 imply for the map  
                                    H^M_z_1,…,z_n(x)=(d_M(x,z_j))_j=1^n
                                
                                 that there are uniform constants ρ_*>0 and c_*>0 such that we have
                                 
                                 |H^M_z_1,…,z_n(x)-H^M_z_1,…,z_n(x')|≥ c_*d_M(x,x'),  for all  x,x'∈ B_M(x_0,ρ_*).
                                 
                                 Let us now choose ξ^0_j∈ S_x_0M, j=1,…,n 
                                 such that s_1ξ_j^0∈𝔹 satisfy
                                 ξ^0_j-ξ_0<c_*/(2s_1),
                                   ξ^0_j-ξ^0_k>c_*/(4s_1)  for j≠k.
                                  Let 
                                  z_j^0=exp_x_0(s_1ξ^0_j). As 
                                  dF^-1≤ C in F(𝔹), there is a uniform constant 
                                  τ_*∈ (0,r_0/100)
                                 such that if z̃_j∈ B(p, r_0/4), j=1,…,n satisfy d(z̃_j,z_j^0)<τ_* then 
                                 there are ξ̃_j∈ S_x_0M and t̃_j>0 such 
                                 that z̃_j=exp_x_0(t̃_jξ̃_j) and
                                 
                                    |ξ̃_j-ξ_j|<r_*/(8s_1)  and   |t̃_j-s_1|<r_*/8.
                                 
                                 Then ξ̃_j and t̃_j  satisfy (<ref>).
                                 Thus,  (<ref>) implies that the map H^M_z̃_1,…,z̃_n(x)
                                 satisfies (<ref>).
                                
                                
                                
                                This implies that if {ẑ_i∈ B_M(p, r_0/4), i=1,2,…,i_M} is any τ_*-net in B_M(p, r_0/4) then for all j=1,2,…,n there are i_j∈{1,2,…,i_M} such that d_M(ẑ_i_j,z^0_j)≤τ_*.
                                Then the above implies that 
                                 for H^M_ẑ_i_1,…,ẑ_i_n(x)
                                 satisfies (<ref>).
                                
                                  Observe that above (x,sξ)∈𝔹, so that 
                                  d_M(x,x_0)<r_*<r_0/100. Moreover, d(x_0, p_1) ≥ r_0/5, 
                                  z_j^0∈ B(p_1, r_0/100),
                                  d_M(ẑ_i_j,z^0_j)≤τ_* <r_0/100
                                 
                                
                                   yield d_M(x,ẑ_i_j)
                                 ≥  r_0/5-3r_0/100
                                 >r_0/8.
                                  
                                
                                Note that above c_*, τ_* and ρ_* are uniform  constants and the estimate 
                                 (<ref>) is valid for some   points
                                 ẑ_i_j in any τ_*-net ẑ_i in B(p, r_0/4), that satisfy d_M(x,ẑ_i_j) >r_0/8,
                                  and any  (M,g,p)∈_ n.
                                 
                                
                                This proves (<ref>) and (<ref>) in claim (i).
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                Next we consider the claim (ii).
                                Let us show that there are  h_1>0 and τ_1>0 such that for any
                                 (M,g,p)∈_ n  and any maximal τ_1-separeted set 
                                 {z_1,…,z_L-1}⊂ B(g,r_0/4) we have
                                 sup_x,y∈ M, x≠y(sup_j_1,…,j_n|(d_M(x,z_j_i))_j=1^n-(d_M(y,z_j_i))_j=1^n|_^n/d_M(x,y))≥ h_1,
                                 
                                where the supremum is taken over all 1≤ j_1<j_2<…<j_n≤ L-1.
                                
                                Assume the opposite. Then for all k∈_+ there are h_k>0, (M_k,g_k,p_k)∈_ n and 1/k-nets
                                {z_j^k: j=1,2,…,L_k}⊂ B(p_k,r_0/4) and points x_k,y_k∈ M_k, x_k ≠ y_k  
                                so that h_k→ 0 and 
                                 sup_x,y∈ M_k, x≠y(sup_j_1,…,j_n|(d_M_k(x_k,z^k_j_i))_i=1^n-(d_M_k(y_k,z^k_j_i))_i=1^n|_^n/d_M_k(x_k,y_k))<h_k.
                                 
                                
                                
                                
                                Using compactness arguments for _ n and choosing 
                                a suitable subsequence of the manifolds (M_k,g_k,p_k) we can assume that 
                                (M_k,g_k,p_k) → (M,g,p) in the  Lipschitz-topology. Then  there are diffemorphisms
                                 F_k:M_k → M such that F_k(p_k)→ p and Lip(F_k)→ 1 and Lip(F_k^-1)→ 1.
                                Moreover,  we can assume that   F_k(x_k)→ x and F_k(y_k) → y  in M and, after using the Cantor diagonalization procedure, we can assume that there are limits lim_k→∞ F_k(z_j^k)=  z_j in M, for all 
                                j=1,2,  …. 
                                Next,  using (<ref>), we see that d_M_k(x_k,y_k) → d_M(x, y),
                                d_M_k(x_k, z_j^k) → d_M(x, z_j) and d_M_k(y_k, z_j^k) → d_M(y, z_j). Also
                                {z_j}_j=1^∞ is dense in B_M(p, r_0/4). Therefore, d_M(x, z)=d_m(y, z) for
                                all z ∈ B_M(p, r_0/4).
                                Then <cit.> (see also <cit.>), 
                                implies that x=y. 
                                
                                Let k  be so large that 1/k<τ_*/2, and
                                      d_M(F_k(p_k),p)<τ_*/2,  Lip(F_k)≤ 2,   Lip
                                (F_k^-1)≤ 2,
                                
                                      
                                d_M(F_k(x_k), x)<ρ_*/4,  
                                d_M(F_k(y_k), y)<ρ_*/4,   h_k<c_*.
                                
                                 As x=y, these imply
                                d_M(F_k(x_k), F_k(y_k))<ρ_*/2 and henced_M_k(x_k,y_k)<ρ_*.
                                As 1/k<τ_*/2, 
                                the points z^k_j,j=1,…,L-1 form a τ_*-net in B_M_k(p_k,r_0/4).
                                Then the inequality (<ref>) for x_k and y_k, 
                                with h_k<c_*, is in contradiction with 
                                the fact that   there is a subset of n of the points in  τ_*-net  
                                 z^k_j,j=1,…,L_k for which 
                                (<ref>) holds. 
                                 This proves (<ref>) with some uniform constants τ_1  and h_1.
                                
                                 We observe that a maximal τ_1-separated subset in  the ball B(p,r_0/4)
                                has at most C_*=(B(p,r_0/4))/_n,R(B(x,τ_1)) points,
                                where _n,R(B(x,τ_1))  is the volume of the ball of radius τ_1
                                on the n-dimensional sphere having  constant curvature R. Hence 
                                we see that the number  of
                                 points for a maximal τ_1-separated subset in B(p,r_0/4) is bounded by a uniform constant  C_*. Thus we can choose L to be the integer part of C_* and τ_0=min(τ_*,τ_1), which makes L and τ_0 uniform constants.
                                
                                
                                As the number L-1  of points in the  τ_1-nets we consider is bounded by a uniform constant, we see that
                                (<ref>) is valid with =h_1^-1+L.
                                These prove the claims (i) and (ii).
                                
                                
                                The above considerations 
                                
                                bring about the following result.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                 There exist uniform constant >0 and 
                                 uniform constant N_F∈ℤ_+ (that is, 
                                
                                 and the integer N_F depend only on  n, R, D, i_0, r_0)
                                
                                 such that
                                
                                 (i)  Let σ∈ (0,τ_0]. Then  any maximal  σ-separated set
                                 x_1, …, x_N(σ) in M
                                 is such that the number of its elements fulfills the bound
                                
                                
                                 N(σ) ≤Ñ(σ)= σ^-n.
                                
                                
                                Moreover,  the balls B(x_k, 4 σ)  satisfy the finite intersection property with at most  N_F intersections, that is, any point x∈ M  belongs to at most N_F
                                balls B(x_k, 4 σ).
                                 (ii) Let  σ∈ (0,τ_0]. Then  any maximal  σ-separated set
                                  z_1, …, z_N_1(σ)
                                 in B(p, r_0/4) 
                                  is such that the number of its elements fulfills the bound
                                
                                N_1(σ) ≤Ñ(σ),
                                and the balls  B(z_k, 4 σ) satisfy the finite intersection property with at most  N_F intersections. Proof.  It remains to prove the finite intersection property. It follows from (<ref>)
                                if we take into the account that
                                B(x_k, 4σ) ∩ B(x_j, 4σ)=∅ if d(x_k, x_j) ≥ 9 σ and
                                B(x_k, σ/2) ∩ B(x_j, σ/2) =∅.
                                
                                
                                § WAVE EQUATION: STABILITY FOR THE UNIQUE CONTINUATION
                                
                                
                                Consider the initial-value problem
                                for the  wave
                                equation
                                
                                    _t^2 w -Δ_g w=0 in  M×,  
                                          w|_t=0=v,   
                                    w_t|_t=0=0,
                                
                                on (M,g,p)  ∈_ n(R, D, i_0) and denote its solution by w=W(v).
                                Our main interest lies in the case when v ∈^̋s_Λ(M), Λ >0,
                                ^̋s_Λ(M)={v∈ H^s(M): v_H^s(M)≤Λ}
                                and we assume in the following that 
                                
                                3/2 < s <2
                                 and denote ^̋0_Λ(M):={v∈ L^2(M): v_L^2(M)≤Λ}.
                                Using the Fourier decomposition we show that, if v ∈ H^s(M), then
                                w_H^s(M× [-T,T])≤ 6 √(T)v_H^s(M)≤v_H^s(M),    T <2 D,
                                
                                where
                                
                                = 6 √(1+D^2).
                                
                                
                                
                                
                                Associated to the wave operator are the double cones of influence. To define these, let V⊂ M be open, T ∈_+.
                                Denote by
                                
                                    Γ(V, T):=V× (-T,T).
                                
                                Then the double cone of influence is given by
                                
                                 
                                    Σ(V, T):={(x,t); d(x,V)+|t| < T}.
                                
                                
                                
                                By Tataru's uniqueness theorem <cit.>, <cit.>, if u is a solution to  (<ref>) 
                                in M × (-T, T),
                                which satisfies u=0 in Γ(V, T), then u=0 in Σ(V,  T).
                                However, for our purposes we need an explicit estimate which follows from Theorem 3.3 in <cit.>.
                                 To formulate the results we introduce, for
                                   0< ≤r_0/32,   r_0/8≤ T <2D,
                                with r_0 fulfilling (<ref>) and
                                z∈ M, the  domains 
                                      Γ=Γ(z, T)=B(z,r_0/16) × (-T+r_0/16, T-r_0/16),
                                
                                   =   (z,γ,T)={(x,t):  (T-d(x,z))^2-t^2 ≥γ^2,  |t| < T-r_0/16},
                                
                                      Ω(T)= M × (-T+r_0/16,   T-r_0/16).
                                 Also, let for b∈,
                                 
                                Σ(z,bγ,T) = {(x,t)∈ M×: |t|≤ T-r_0/16 , |t| ≤  T-bγ - d_g(x,z)}
                                 be the “domain of influence” corresponding to the cylinder Γ(z, T).
                                 Observe that 
                                 Σ(z,γ,T) ⊂(z,γ,T)  ⊂Σ(z,0,T). In the following we formulate the stability results for the unique continuation in <cit.>. We note 
                                that similar results have been obtained by Luc Robbiano in <cit.>   with θ=1, but with a  loss
                                 in the domain of dependence and later by  C. Laurent and M. Leautard
                                 in <cit.> with θ=1, but without an explicit calculation
                                 of the constants in the  domain of dependence.
                                
                                
                                
                                
                                Let (M,g,p) ∈_ n(R, D, i_0).
                                Let P=P(x,D)=^2_t-Δ_g  be the wave operator associated with  M.
                                Assume that w(x,t)=0 for all (x,t)∈Γ.
                                Then, for any
                                0 <θ<1, there is c_206(,θ)≥ 1,
                                 depending only on n, R, D, i_0,r_0,θ and γ
                                 such that the following stability estimate holds true:
                                
                                    w_L^2((z, , T))≤ 
                                    c_206(,θ)w_H^1(Ω(T))/(ln(1 + w_H^1(Ω(T))/Pw_L^2(Ω(T))))^θ ,
                                
                                where c_206(,θ) is such that
                                
                                
                                c_206(,θ) = c_205 ( θ)exp( ^-c_200),   c_200=58(n+1)+1,
                                
                                and  c_205( θ)≥ 1 depends on 
                                
                                θ, n, R, D, i_0, r_0.
                                Moreover, for any 0 ≤ m ≤ 1,
                                
                                    w_H^1-m((z, ,T))≤ 
                                    c_206(,θ)^mw_H^1((Ω(T))/(ln(1 + w_H^1((Ω(T))/Pw_L^2((Ω(T))))^θ m .
                                 Proof.
                                Theorem <ref> follows from Theorem 3.3  in <cit.>
                                with ℓ=r_0/16 and
                                (z,γ,T) = S(z, r_0/16, T, ).
                                Using that w=0 in Γ, the domain Λ in the final equation of Theorem 3.3
                                can be changed into (z, ,T).
                                Moreover, for θ <1, the function f_θ(a,   b), a, b>0,
                                f_θ(a,b)=
                                a (ln(1+a/b))^-θ,
                                
                                increases when either a or b increases. Thus,
                                we can change w_H^1(Ω_1) and P w_L^2(Ω_1) in Theorem 3.3
                                to w_H^1(Ω(T)) and P w_L^2(Ω(T)). 
                                 Note that, although the results in <cit.> are formulated for M ⊂^n, 
                                
                                
                                they can be easily reformulated for an arbitrary compact Riemannian manifold which
                                possess C^5-smooth covering by coordinate systems with C^4-smooth metric tensors.
                                
                                 To consider parameters  (<ref>)  (see the Appendix for details),
                                we will fix the value of θ  to beθ = 1/2,
                                
                                 for simplicity. In the general case, we write c_205 ( θ) as θ-dependent.
                                
                                We recall that the constants
                                in <cit.> (see (3.1)) explicitly depend on parameter >1 such that
                                ^-1|ξ|^2 ≤ g^jk(x)ξ_jξ_k≤|ξ|^2,
                                  g^jk(x)_C^4(M)≤.
                                Using harmonic coordinates in balls of radius r^(har), this condition is fulfilled  due to (<ref>), which also implies d_g(x,z) ∈ C^3. 
                                
                                
                                
                                Our main interest will be an estimate for v(·)=w(0, ·) in (<ref>) in the domain
                                B(z, T-2). 
                                
                                
                                
                                
                                 
                                Assume (<ref>) and let 
                                θ∈ [1/2,1). Also, let 
                                Λ_1>0 and_2 ∈ (0,Λ_1]
                                and
                                v ∈^̋s_Λ_1(M).
                                Denote by
                                w=W(v)  the solution to initial-value problem (<ref>) and assume that,
                                w_L^2(B(z, r_0/16+) × (-T +r_0/16,  T-r_0/16))≤_2.
                                
                                Then, calling β=θ^2/2 and defining _1
                                :=ℰ_1(_2; θ, γ,  Λ_1), we get
                                v_L^2(B(z,T-2))   ≤   _1
                                
                                where, for c_202=c_202(θ, ),  and (θ)
                                depending only on θ  and n,R,D,i_0,r_0ℰ_1(_2; θ, γ,  Λ_1)    =   
                                 c_202 Λ_1/^(2-θ/2)(ln[1+ Λ_1^(s-1)/s_2^-(s-1)/s]  )^β,  
                                
                                c_202   =   exp(^-(c_200 θ/2)),
                                
                                   =   (θ) ≥ 1.
                                 Proof. 
                                Let the cut-off function η(x)∈ C^2_0(B(z, r_0/16+/2)) be equal to
                                one in B(z,r_0/16) and η_C^i(M)≤ Cγ^-i,  i=0, 1,2. Then
                                w_η(x,t)=(1-η(x)) w(x,t) vanishes in Γ and we have
                                (_t^2- Δ) w_η(x,t)=F, where
                                
                                F(x,t)   =   (Δ_g η(x) ) w(x,t)+ 2g(∇η(x),∇_x w(x,t)) 
                                   =   (Δ_g η(x) ) (η̃(x) w(x,t))+ 2g(∇η(x),∇_x ( η̃(x) w(x,t))) := F_1+F_2
                                
                                Here η̃(x)∈ C^2_0(B(z, r_0/16+)) is equal to
                                one in B(z, r_0/16+ γ/2) and η̃_C^i(M)≤ Cγ^-i,  i=0, 1,2.
                                Clearly, by hypothesis
                                
                                
                                    F_1_L^2(M × (-T +r_0/16,  T-r_0/16))≤  C ^-2_2.
                                 
                                To estimate
                                F_2, observe that,η̃w_H^s(M × (-T+r_0/16, T-r_0/16))≤ C ^-sΛ_1,
                                where we have also used (<ref>). Since
                                
                                η̃w_L^2(M × (-T+r_0/16, T-r_0/16))≤_2,
                                
                                by interpolation arguments,
                                we get
                                 η̃w_H^1(M × (-T+r_0/16, T-r_0/16))≤ C ^-1Λ_1^1/s_2^1-1/s
                                Since supp(∇η) ∩supp(∇η̃) =∅, this implies
                                
                                
                                
                                F_2_L^2(M × (-T +r_0/16,  T-r_0/16))≤  C ^-1Λ_1^1/s_2^1-1/s,
                                F_L^2(M × (-T+r_0/16, T-r_0/16))≤ C ^-2Λ_1^1/s_2^1-1/s,
                                
                                where we used _2 ≤Λ_1. As s>1, we have
                                  w_η_H^1(M × (-T+r_0/16, T-r_0/16))≤ C ^-1Λ_1.
                                
                                
                                Using growth properties of the function f_θ
                                of form (<ref>), it follows from Theorem <ref> that
                                 w_η_H^1-θ/2()≤
                                C  c_206(,θ)^θ/2^-1Λ_1/(ln[1+ Λ_1^(s-1)/s_2^-(s-1)/s])^β.
                                
                                Now observe that by the trace-theorem, for any >1/2 there exists =()
                                such that, for r ≥ r_0/16,  z ∈ M:
                                w( ,0)_L^2(B(z, r))≤ γ^-w_H^( B(z, r)× (-,)),
                                w( ,0)_L^2(B(z, T-2 ))≤ γ^-w_H^ ((z,γ,T)).
                                
                                It follows from  (<ref>) with =1-θ/2 and (<ref>)
                                that,
                                
                                
                                    w_η(,0)_L^2(B(z,T-2γ))≤ C  c_206(,θ)^θ/2Λ_1/^2-θ/2(ln[1+ Λ_1^(s-1)/s_2^-(s-1)/s])^β.
                                
                                
                                Next define =(1-β) s +β >1/2. Then by  interpolation,
                                η w  _H^( B(z, r)× (-,))≤
                                c_201 η w^/s_H^s( B(z, r)× (-,)) η w^(s-)/s_L^2(B(z, r)× (-,)).
                                
                                Using the fact that supp(η) ⊂ B(z, r_0/16 +),
                                we can apply (<ref>) with r=r_0/16 +, the previous inequality and (<ref>), to obtain
                                η(·) w(·, 0)  _L^2(B(z,T-2γ))≤^- c_201(Λ_1)^/sϵ_2^β(s-1)/s
                                ≤ ^β- c_201^/sΛ_1(ln[1+ Λ_1^(s-1)/s_2^-(s-1)/s])^-β
                                
                                .
                                
                                Here at the last step we
                                
                                use the fact that
                                
                                X ≥ln(1+X) for X >0, with X = γΛ_1^(s-1)/s_2^-(s-1)/s.
                                Recall  that v(x)=w_η(x,0)+η(x) w(x,0).
                                Comparing (<ref>) and (<ref>), we obtain
                                equation (<ref>).
                                The  coefficient c_202 defined in (<ref>)  fulfills the inequality
                                
                                    c_202≥ C  c_206(,θ)^θ/2^θ/2-2
                                    +  c_201 ^(1-β)+β/s^(β-1)s,
                                
                                by using (<ref>) and a proper multiplicative coefficient  independent on γ.
                                
                                
                                § COMPUTATION OF THE PROJECTION
                                
                                
                                 §.§ Domains of influence
                                 Let (M,g,p)∈_ n(R, D, i_0). By Proposition <ref>, we can choose L-1 points z_j, j=1,2,…,L-1 that
                                form a τ_0-net in B_M(p,r_0/4). Here, L is bounded by a uniform constant. In  Lemma <ref> we showed that for any σ there are 
                                N_1(σ) points, that we enumerate as  z_L, …, z_ L-1+N_1(σ),
                                which form a maximal σ-separated net in B(p, r_0/4) 
                                and the balls  B(z_k, 4 σ), k=L,…, L-1+N_1(σ), satisfy the  finite intersection property with at most  N_F intersections.
                                In this section we consider arbitrary σ, which value will be specified later, and points  z_ℓ,  ℓ=L, …, L-1+ N_1(σ) that satisfy the conditions
                                of Lemma <ref>. Also, below is 3/2<s<2.
                                
                                
                                
                                Our  next goal is to approximately construct
                                the values of the distance functions from a variable point x ∈ M to all points
                                z_ℓ∈ B(p,r_0/4), ℓ=1,2, …,   L-1+N_1(σ), defined in Lemma <ref>.
                                
                                The main step is to
                                approximately compute the Fourier coefficients of the  functions of form
                                χ_Ω (x)v(x), where χ_Ω(x) are the characteristic functions of some special subdomains
                                Ω⊂ M
                                and v(x) has  a finite Fourier expansion. These subdomains Ω are defined using distances to
                                L points
                                {z_1, …, z_L-1, z_i }, where i ∈{ L, …,   L-1+N_1(σ)} is arbitrary.
                                For  i ∈{ L,L+1, …,   L-1+N_1(σ)},
                                let
                                K_i={1,2,…,L-1}∪{i}
                                and define 𝒜^(i) to be the set of those =(_ℓ)_ℓ=1^ L-1+N_1(σ)∈^ L-1+N_1(σ), such that 
                                
                                
                                      r_0/8≤_ℓ≤ 2D, if ℓ∈ K_i,
                                      _ℓ=0, if ℓ∉K_i.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                Below, we will assume that
                                γ≤σ.
                                
                                We
                                
                                 denote 
                                      Γ̃(z, T)=B(z,r_0/16+γ) × (-T+r_0/16, T-r_0/16).
                                
                                
                                
                                
                                Next we fix for a while the index i ∈{ L, …,   L-1+N_1(σ)}.
                                To construct subdomains Ω, we start with
                                 observation sets 
                                Γ̃(), ∈𝒜^(i),
                                Γ̃()=
                                ⋃_ℓ∈ K_iΓ̃(z_ℓ,  _ℓ).
                                
                                At last, for b ∈, we define
                                
                                M(, b )=⋃_ℓ∈ K_i
                                 B_M(z_ℓ, _ℓ+b).
                                 Then the corresponding domains of stable unique continuation are 
                                      ()= ⋃_ℓ∈ K_i(z_ℓ,  , _ℓ),  (,bγ)= ⋃_ℓ∈ K_i(z_ℓ,  , _ℓ+bγ),
                                
                                
                                
                                
                                 
                                 and the corresponding double cones of influences are given by
                                         Σ()= ⋃_ℓ∈ K_iΣ(z_ℓ,  , _ℓ),  Σ(,bγ)= ⋃_ℓ∈ K_iΣ(z_ℓ,  , _ℓ+bγ),
                                 
                                
                                
                                We have the following volume estimate.
                                
                                a) Let ∈𝒜^(i),  i= L,…,  L-1+N_1(σ) and
                                
                                A=A(, ) = {x∈ M: d(x, M(,   3))≤ 5}.
                                
                                Then, there is a uniform  constant >0,
                                
                                 depending only on n, R, D, i_0 and r_0,
                                
                                 such
                                that
                                
                                    (A)≤ L .
                                b) Consequently, by defining b(s), for 3/2 <s <2,  as 
                                b(s) = 1/2,  n=2, 3 and b(s) = s/n,   n ≥ 4,
                                we see that there is a uniform constant (s),
                                
                                 depending only on  s, n, R, D, i_0 and r_0,
                                
                                 such that  
                                χ_ B(z_ℓ, _ℓ+ 8) ∖ B(z_ℓ, _ℓ- 2 ) v_L^2(M)≤(s)  ^b(s)v_H^s(M).
                                 Proof.a) Let x ∈ A. Then, for some 
                                ℓ∈ K_i,
                                
                                x ∈ B(z_ℓ, _ℓ+ 8) ∖ B(z_ℓ, _ℓ-  2 ).
                                
                                Since d exp_z_ℓ|_ v is uniformly bounded on _ n(R, D, i_0) for
                                v∈ T_z_ℓM, | v| ≤ 2D,
                                ( B(z_ℓ, _ℓ+ 8) ∖ B(z_ℓ, _ℓ-  2 ) )
                                ≤ C, for all ℓ∈ K_i. 
                                
                                b)  Similar to part a), we have
                                (B(z_ℓ, _ℓ+ 8) ∖ B(z_ℓ, _ℓ- 2) )≤ c .
                                
                                Together with the Hölder inequality and  the Sobolev embedding H^s(M) → L^q(M), 1/q =1/2 -s/n, 
                                (or C^0(M) for n= 2, 3), this implies (<ref>).
                                Note that (s) is a uniform constant as the embedding can be done in
                                harmonic coordinates defined in balls with uniform radius.
                                
                                
                                  §.§.§ Cut-off estimates and finite dimensional projections
                                 Let us apply  Lemma <ref>, with  instead of σ,
                                to obtain points x_ℓ∈ M, ℓ=1,2, …, N()
                                such that the balls B(x_ℓ, 2 ), ℓ=1,2, …, N() are a covering of M.
                                
                                Let ψ_ℓ:M→_+,  ψ_ℓ∈ C^6_*(M)
                                be  in harmonic coordinates a partition of unity for the covering B(x_ℓ, 2 ) that satisfy
                                ψ_ℓ_C^k,β(M)   ≤    c_k,β^-(k+β),
                                   k=0, 1, 2,   0≤β <1;
                                 
                                      (ψ_ℓ) ⊂ B(x_ℓ, 2),  ∑_ℓ=1^N()ψ_ℓ(x)=1.
                                Below, we use Λ_s≥ 1. For 3/2<s<2 there is(s)≥  1, in (<ref>) such that, for any
                                u∈^̋s_Λ_s(M),  i ∈{L, …,  L-1+N_1(σ)}
                                and ∈𝒜^(i), the following holds true:
                                There 
                                
                                
                                
                                
                                      u_∈^̋s_1/4(s;) Λ_s(M) ∩^̋0_Λ_s(M),  
                                u_(x)=0,  if x∈ M(, ), 
                                       u_(x)=u(x),  if x∈ M ∖ M(,   7),
                                
                                where
                                (s, )=  (s) ^-s.
                                 Proof.
                                Define
                                
                                u_(x)=
                                Ψ(x)u(x),  Ψ(x)=
                                ∑_supp(ψ_ℓ)∩ M(,  3)=∅ψ_ℓ(x).
                                 For a general w∈ H^2(M) we have 
                                
                                the following estimate in Sobolev spaces with 3/2<s<2Ψ w_H^s(M)≤ CΨ_C^s(M)w_H^s(M)≤ C m  c_2,0γ^sw_H^s(M),
                                
                                where m is the number of elements in the set {ℓ: supp(ψ_ℓ)∩ M(,  3)=∅} satisfying  m≤ N(). 
                                
                                
                                
                                Thus the existence of  (s) such that the claim holds follows then
                                from the finite intersection property
                                of B(x_ℓ, 2 ), see Lemma <ref>, and estimates 
                                (<ref>).
                                
                                
                                 §.§ Unique continuation for approximate projections
                                
                                
                                
                                Corollary <ref>  implies the following result.
                                
                                Note that the notations _2 and ℰ_2 are introduced in order to distinguish _2 from its upper bound ℰ_2, written as an expression dependent on _1.
                                 
                                
                                
                                Later, in formula (<ref>) we set  _1  to have
                                a specific value and substitute it in the expression ℰ_2( _1/4L; θ, ,  Λ_s)  of formula (<ref>)
                                to obtain a specific value for _2. 
                                Assume that v satisfies
                                v_H^s(M)≤ (s, ) Λ_s  and  v_L^2(M)≤Λ_s, 
                                
                                 with (s, ) defined in (<ref>), and assume (<ref>).
                                 
                                Let _1 <Λ_s and _2 ≤_2(  _1/4L;θ, , Λ_s)    where
                                
                                ℰ_2( _1/4L; θ, ,  Λ_s)
                                 = Λ_s   ^s/(s-1)/(exp[( 
                                 Λ_s 4L _1^-1γ^-(2-θ/2)   exp( γ^-c_200) )^1/β])^s/(s-1)  
                                Let w= W(v) satisfy 
                                w_L^2(Γ̃(z_ℓ,  _ℓ))≤_2
                                
                                
                                
                                on the domain (<ref>).
                                Then, for ℓ∈ K_i,
                                w(0, ·)_L^2(B(z_ℓ, _ℓ- 2))≤_1/4L, 
                                  w(0, ·)_L^2(M(,- 2))≤1/4_1.
                                
                                
                                 Proof.  
                                
                                From a small modification of the proof of Corollary  <ref> we still can obtain the estimate (<ref>) in the following way.
                                
                                The main point is to replace the initial condition v_H^s(M)≤Λ_1 with (<ref>).
                                We then deduce the corresponding estimate for the solution w=W(v) of the wave equation, with T=_ℓ and z=z_ℓ,
                                w_H^s(M×[-T,T])≤ C(s, ) Λ_s  and  w_L^2(M×[-T,T])≤  CΛ_s, 
                                 Let η and  η be the smooth localizers defined in the proof of Corollary  <ref>. 
                                Calling again w_η = (1-η(x))w and using the definition of  in (<ref>) we get, 
                                η w_H^s(M× (-T+r_0/16,T-r_0/16))≤ C γ^-sΛ_s, 
                                ηw_H^s(M× (-T+r_0/16,T-r_0/16))≤ C γ^-sΛ_s, 
                                 w_η_H^s(M× (-T+r_0/16,T-r_0/16))≤ C γ^-sΛ_s, 
                                
                                and the intermediate H^m norms follow  by interpolation.  
                                
                                Here the constant C is
                                
                                dependent of
                                
                                c_3(s) and independent of . Consequently,
                                η̃w_H^1(M × (-T+r_0/16, T-r_0/16))≤ C ^-1Λ_s^1/s_2^1-1/s,
                                
                                F_L^2(M × (-T+r_0/16, T-r_0/16))≤ C ^-2Λ_s^1/s_2^1-1/s.
                                
                                Using growth properties of the function f_θ
                                we get (<ref>).
                                Also (<ref>) still holds.
                                Therefore we obtain (<ref>), where the new constant  in (<ref>) now depends on c_3(s).
                                
                                
                                
                                
                                
                                Next we observe that formula (<ref>) implies that when 
                                _1=4Lℰ_1(_2; θ, γ, Λ_s), we have
                                _2 = Λ_s γ^s/(s-1)/( exp[ (  Λ_s 4L 
                                _1^-1γ^-(2-θ/2)exp(^-(c_200 θ/2))
                                
                                  )^1/β] -1 )^s/(s-1)  ,
                                
                                and ℰ_2 is defined  by removing -1 from the denominator of the expression above, and by replacing exp(^-(c_200 θ/2)) with exp(^-c_200).
                                This is done to simplify the calculations of the paper.
                                The relation (<ref>) follows by imposing on _2 the ℰ_2-bound.
                                
                                
                                Under the conditions of the Corollary and from the growth properties of ℰ_2(_1) it follows that
                                _2 ≤ℰ_2( _1/4L; θ, ,  Λ_s) ≤_1/4L,   _1 ∈ (0, Λ_s].
                                
                                
                                 §.§ Approximate projections
                                
                                Let _0,_1,_2
                                satisfy  
                                _0 ≤Λ_s/10,
                                
                                
                                   _1=_0^2/10 Λ_s,    _2 =_2(  _1/4L, θ, , Λ_s ).
                                
                                
                                  §.§.§ Finite data with and without errors
                                
                                Below we will use several parameters, and for the sake of clarity of presentation,
                                we have gathered these parameters in this subsection and tell how those will be used.
                                
                                 Below, 
                                we will use ∈_+ satisfying
                                   ≥   (_2/8;  , Λ_s),
                                
                                
                                where
                                ( _*;  , Λ_s)=
                                ^-n(Λ_s/_*)^n/s  and  (s)= (s)^n/s^n/2(+1)^n/s.
                                
                                We also use 
                                ∈_+ satisfying 
                                ≤≤  2^n/2()^n
                                Moreover,
                                we use 
                                δ   ≤   δ_0(_2, , , Λ_s)=
                                 1/ _2/Λ_s,
                                
                                where 
                                 =min(^-1 , (1+ 2
                                )^-1/2/100(1+D)^3/2L),
                                and J satisfying
                                
                                ( ^-1δ^-1)^n/2≤ J≤   (2δ^-1)^n/2,
                                
                                cf. Remark <ref>.
                                Note that (<ref>) implies that λ_J≥δ^-1, see 
                                 Def. <ref> (ii) and (<ref>).
                                
                                The use of the above parameters are the following.
                                We will assume that we are given  the ball (B_e(r_0), g^a) and the pairs
                                {(λ_j^a, φ_j^a|_B_e(r_0)) ;  j=0,1,2,…,J}. 
                                
                                
                                We assume
                                that these data are δ-close to
                                FISD of some manifold (M,g,p)∈_ n,
                                that is, the ball (B_e(r_0), g) and 
                                {(λ_j, φ_j|_B_e(r_0)) ;  j=0,1,2,…,J},
                                where the error size parameter δ satisfies (<ref>).
                                
                                
                                
                                
                                We are going to formulate a minimizaton algorithm that will be used to compute 
                                volumes of the sets (<ref>).
                                
                                We consider this minimizaton algorithm in the two cases, in the case when we have FISD without errors 
                                and the case when we have it with  errors.
                                
                                
                                
                                
                                
                                
                                
                                
                                As we have finite data, we need to consider the projection of the solution
                                of the wave equation to finitely many eigenvectors, and we choose 
                                so that it is enough to use  eigenvectors.
                                This requires that we have the data  (λ_j, φ_j|_B_e(r_0)) with j=0,1,2,…,.
                                However, to consider minimization algorithms both for FISD with and without errors, we need to 
                                increase the amount of data and we will consider
                                 (λ_j, φ_j|_B_e(r_0)) with j=0,1,2,…,,
                                where  is chosen as follows:
                                
                                
                                In Definition <ref>,
                                there are intervals I_p⊂,  p=0,1, …, P covering the spectrum of M in 
                                [0, δ^-1+ δ]
                                each  I_p containing a cluster of
                                 n_p eigenvalues λ_j and approximate eigenvalues λ_j^a. 
                                To consider these clusters of eigenvalues, let
                                P_0 be the smallest integer P_0≤ P  such that
                                 {λ_0,λ_1,…, λ_}⊂⋃_p=0^P_0 I_p
                                
                                and then choose  such that  ≤≤ J and
                                 
                                j≤λ_j∈⋃_p=0^P_0 I_p,  
                                j> λ_j∉⋃_p=0^P_0 I_p.
                                
                                We note that this happens with some   satisfying (<ref>). 
                                We also observe that as  δ  satisfies (<ref>)  and J satisfies (<ref>),
                                and as Λ_s≥ 1, _2<1 and n≥ 2, we have
                                
                                J≥ J_0(δ) =(^-1δ^-1)^n/2≥.
                                
                                
                                  §.§.§ Minimisation with FISD without errors
                                 
                                
                                Let _0,_1, _2
                                satisfy (<ref>). 
                                There is _0(_0;  s, Λ_s) depending only on _0,s,Λ_s, n, R, D, i_0 and r_0,
                                
                                 
                                
                                
                                with the following properties:
                                Let ≤_0(_0;  s, Λ_s). 
                                Assume that
                                 satisfies (<ref>) and  satisfies (<ref>), and
                                 
                                u(x)=∑_j=0^ a_jφ_j(x) ∈^̋s_Λ_s(M),
                                
                                
                                Let i ∈{L, …,   L-1+N_1(σ)} and
                                ∈𝒜^(i).
                                
                                Moreover,  assume that we are given
                                (B(p, r_0),g|_B(p, r_0)),  ((λ_j, φ_j|_B(p, r_0)))_j=0^  and    (a_j)_j=0^.
                                  The data (<ref>) determine the set ^*_ m and the function Ł_ a:C^*_ m→, 
                                
                                defined in (<ref>) and (<ref>), 
                                
                                for which the 
                                
                                minimizer of L_ a in ^*_ m
                                
                                
                                
                                
                                is  a sequence(d_j)_j=0^=(d_j(α,i))_j=0^∈^ 
                                 such that
                                v(x)=∑_j=0^  d_j^φ_j(x) satisfies
                                
                                
                                v ∈^̋s_ (2 (s, ) Λ_s)(M) ∩^̋0_2 Λ_s(M),   v-χ_M(, -2)u_L^2(M)< _0,
                                
                                
                                
                                The  above bound  _0(_0; s, Λ_s) for γ
                                is defined in (<ref>). 
                                 Note that the sequence (d_j)_j=0^ is not unique and that the theorem states the existence of sequences satisfying (<ref>). The next subsections are devoted to the proof of Theorem <ref>.
                                In sec. <ref>, <ref> and <ref> we keep the index i ∈{L, …,   L-1+N_1(σ)} fixed not referring to this.
                                
                                
                                
                                
                                  §.§.§ Finite dimensional projections
                                
                                
                                
                                Next we introduce some special sets of the finite-dimensional functions.
                                
                                Let b=(b_j)_j=0^∈^(+1) and  ^*( b) be its Fourier coimage
                                ^*( b)=∑_j=0^ b_jφ_j ∈ L^2(M).
                                
                                For a_1,  a_2>0  the class of Fourier coefficients _,s(a_1, a_2) is defined as
                                _,s(a_1,a_2):={ b∈^(+1);   ∑_j=0^(1+λ_j^2)^s|b_j|^2≤ a_1^2,  ∑_j=0^ |b_j|^2≤ a_2^2}.
                                
                                For w=W(v) being the solution to the  problem (<ref>)
                                and
                                b∈^(+1),  we denote
                                𝒲( b)=W(^*( b)) ∈ C(;L^2(M))
                                
                                and, for any _* >0, ∈𝒜^i, we denote
                                      _, s( _*;  a_1,  a_2, α) 
                                
                                      =
                                { b∈_ ,s(a_1, a_2) : W(^*( b))_L^2(Γ̃(z_ℓ, _ℓ ))≤_*,  ∀ℓ∈ K_i}.
                                 (i) Let v∈^̋s_ (s,   ) Λ_s(M) 
                                and let P_j' be the orthoprojection
                                P_j'v=∑_j=0^j'⟨v|,φ_j_L^2(M) φ_j.
                                Then for any ∈𝒜^(i), _2>0, P_j'v-v_L^2(M)≤1/ 8(+1) _2,   if   j' ≥≥(_2/8;  , Λ_s),
                                
                                see (<ref>) for  and (<ref>) for 
                                ( _2/8;  , Λ_s).
                                (ii) Let u ∈^̋s_Λ_s(M)
                                and u_ be  given by (<ref>).
                                Let  satisfy (<ref>)-(<ref>).
                                Then,
                                
                                 v_=P_ u_α∈^*(_, s( 1/8_2;   1/4(s;) Λ_s,Λ_s,  α)).
                                 Proof.
                                (i) For v = ∑_j=0^∞ b_jφ_j, we have
                                
                                    P_j'v-v^2_L^2(M)= ∑_j >j' |b_j|^2
                                    ≤ 
                                    
                                    
                                     |_j'|^-s(s; )^2Λ_s^2.
                                Here, (s; ) is defined in (<ref>)   and 
                                 (<ref>) with s=0, and these  impy the estimate  (<ref>). (ii) The  finite propagation speed of waves implies,
                                due to u_|_ M(,)=0,  
                                that
                                W( u_)|_Γ̃(z_ℓ, _ℓ)=0.
                                By Lemma <ref> and (<ref>) 
                                
                                    W( v_)_L^2(Γ̃(z_ℓ, _ℓ))≤ W( v_-u_)_L^2(Γ̃(z_ℓ, _ℓ))≤v_ - u__L^2(M)≤1/8_2.
                                
                                Since P__H^s(M)=1  for any s, the claim (i) of the lemma with 
                                j'=, (<ref>)
                                together with  (<ref>) prove (<ref>).
                                 The condition
                                W(^*( b))_L^2(Γ̃(z_ℓ, _ℓ ))≤_*, see (<ref>),
                                is equivalent to
                                  (∑_j=0^  b_jcos(√(λ_jt)) φ_j(x))_L^2(Γ̃(z_ℓ, _ℓ))≤_*,   ℓ∈ K_i.
                                 
                                which  can be directly verified if we know
                                 {(_j, φ_j|_B(p, r_0))}_j=0^. 
                                
                                
                                  §.§.§ Minimisation algorithm
                                
                                
                                
                                Assume that we are  given a=(a_j)_j=1^∈^(+1) and denoteu= ^*( a) ∈^̋s_Λ_s(M).
                                Our next goal is
                                to use  FISD to find a vector 
                                b ∈_, s((s, )Λ_s,  Λ_s)
                                such that ^*( b) is close  to χ_M()^*( a).
                                To achieve this goal  we will use a minimisation method.
                                
                                
                                 
                                Let _0,_1, _2
                                satisfy (<ref>). 
                                
                                
                                
                                
                                Let m∈{1,2,4} be a parameter we will use below, and
                                _ m:=^*(^*),   where  ^*_ m=_, s(  1/2m_2; 1/m(s, )Λ_s, Λ_s, α).
                                
                                (i) A function v ∈_ m
                                is called an _1-minimizer of the minimization problem
                                min_h∈_ mŁ_u(h),  where Ł_u(h)=
                                h-u^2_L^2(M),
                                
                                if v satisfies
                                v-u_L^2(M)≤ J_min(m)+ 5 Λ_s_1  ,   J_min(m):=inf_h∈_ mh-u_L^2(M)^2.
                                
                                (ii)  Equivalently, a vector
                                b=(b_j)_j=0^∈^*_ m  is an _1-minimizer of the minimization problem
                                min_ c∈^*_ mŁ_ a( c),  where Ł_ a( c)=
                                 c- a^2_^(+1),
                                
                                if
                                 b- a_^(+1)^2≤ J_min(m)+ 5 Λ_s _1,   J_min(m):=inf_ c∈^*_ m c- a_^(+1)^2.
                                 
                                Observe that for c ∈_,s(1/m(s, )Λ_s, Λ_s)
                                we can check, using Remark <ref> with _*= _2 /2m, that c∈^*_ m and thus
                                find b which satisfies (<ref>).
                                
                                
                                Next we assume that, in addition to  _2 satisfying (<ref>),  satisfies 
                                
                                      ≤γ_0=_0(_0; s, Λ_s)=(_1/Λ_s)^1/(b(s)),
                                   with  
                                      (s)= 1/( 2 L (s) )^1/(2b(s)),  
                                   _1 = _0^2/10Λ_s,
                                
                                where b(s) and (s) are defined in Lemma <ref>, b).
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                Let  
                                u∈^̋s_Λ_s(M), and let
                                 _0,_1,_2,  , satisfy (<ref>), (<ref>)-(<ref>)
                                and  (<ref>). 
                                
                                
                                (i) For m∈{1,2,4} and all h∈_ m, we have
                                Ł_u(h) ≥u_L^2(M(, -2))^2-2 Λ_s _1
                                + h-u_L^2(M∖ M(, -2))^2.
                                
                                (ii) The function v_ defined by (<ref>),
                                (<ref>)
                                 satisfies
                                 v_∈_ m with m=4 and and
                                Ł_u( v_)
                                ≤u^2_L^2(M(, -2))+2 Λ_s  _1 +4 _1^2.
                                
                                
                                
                                 Note that here v_∈_ 4⊂_ 2⊂_ 1.
                                (iii) For all m∈{1,2,4}, the function v_∈_ m is an _1-minimiser,
                                Ł_u(v_)  ≤ J_min(m)+ 5Λ_s _1 .
                                 (iv) For all m∈{1,2,4}, we have 
                                 |J_min(m)- u^2_L^2(M(, -2))|≤ 2 Λ_s  _1 +4 _1^2.
                                 Proof.
                                (i)
                                We have, for
                                h∈_ m,
                                      h-u_L^2(M)^2= h-u_L^2(M(, -2))^2
                                + h-u_L^2(M∖ M(, -2))^2 
                                   ≥   (u_L^2(M(, -2))-h_L^2(M(, -2)))^2
                                + h-u_L^2(M∖ M(, -2))^2.
                                
                                Since h ∈_ m,  (<ref>),
                                (<ref>) and (<ref>) imply that h_L^2(M(, -2))≤_1.
                                Thus,
                                h-u_L^2(M)^2
                                   ≥   u_L^2(M(,-2))^2-2Λ_s _1 +
                                _1^2
                                + h-u_L^2(M∖ M(,-2))^2.
                                (ii) With u_,  v_ defined by (<ref>) and  (<ref>),
                                v_∈_ 4,      u- v_^2_L^2(M) = u- v_^2_L^2(M(, -2))+
                                 u- v_^2_L^2(M∖ M(, -2 ))
                                      ≤u_L^2(M(, -2))^2 +2  Λ_s _2+
                                 _2^2
                                
                                         +2u- u_^2_L^2(M∖ M(, -2 ))
                                + 2  u_- v_^2_L^2(M∖ M(, -2 )),
                                
                                where we use that u- v_=(u- u_) +( u_ - v_)
                                and
                                v__L^2(M(, -2 ))^2≤_2^2, see Lemma <ref>.
                                Observe, that by (<ref>), (<ref>) and (<ref>),  
                                
                                
                                u-u_^2_L^2(M ∖ M(, -2)) 
                                = u^2_L^2(M(,  7) ∖ M(, -2))≤(s) Λ_s^2 L^2^2b(s)≤1/2_1^2.
                                
                                where (s) is defined in Lemma <ref>, b).
                                Using (<ref>) and (<ref>), we see that 
                                u_-v_^2_L^2(M∖ M(-2 ))≤_2^2.
                                Thus, inequality  (<ref>)  yields that
                                
                                    Ł_u(v_) ≤u^2_L^2(M(, -2))+2 Λ_s  _2+
                                    3  _2^2
                                    +_1^2.
                                
                                 As  _2 ≤_1, see (<ref>), 
                                we get  (<ref>).
                                
                                
                                (iii)
                                The claims (i) and (ii) together with (<ref>) yield that
                                Ł_u(v_)- J_min(m)=Ł_u(v_) -min_h∈_ mŁ_u(h)                     
                                ≤( u^2_L^2(M(, -2))+2 Λ_s _1 +
                                4 _1^2 )-
                                (u_L^2(M(, -2))^2-2 Λ_s _1 )
                                ≤
                                5 Λ_s _1.
                                
                                (iv)
                                 The claim (iv) follows from (i) and (ii).
                                Let m∈{1,2,4} andu∈^̋s_Λ_s(M), 
                                 _0,_1, and _2 satisfy (<ref>) and  (<ref>), satisfies (<ref>)-(<ref>)
                                and  satisfies (<ref>).
                                Let v^*= ∑_j=0^ b_j φ_j be any  _1-minimizer of  the minimization problem (<ref>), with
                                b ∈^*_ m.
                                Then
                                 v^*-χ_(M ∖ M(, -2))u^2_L^2(M)≤_0^2.
                                 Proof.
                                Since v_  satisfies   by (<ref>) and  (<ref>),
                                
                                    v^*-u_L^2(M)^2 ≤ v_-u_L^2(M)^2 +5 Λ_s _1
                                    
                                    ≤u_L^2(M(, -2))^2 +7 Λ_s _1
                                    
                                    + 4 _1^2.
                                
                                Since v^*-u satisfies (<ref>), this inequality implies that
                                 v^*-u_L^2(M∖ M(,-2))^2
                                ≤
                                9 Λ_s _1+ 4 _1^2.
                                
                                Since v^*∈, w^*=W(v^*) satisfies  (<ref>) with
                                 ^*=_2, where _2 satisfies (<ref>) and  (<ref>). 
                                 It then follows from Corollary <ref> 
                                 that
                                 
                                    v^*^2_L^2(M(, -2))≤_1^2.
                                 Due to
                                (<ref>), this inequality
                                 together with  (<ref>), implies
                                 (<ref>).
                                
                                 Proof of Theorem <ref>. Assume that
                                  a:=(a_j)_j=0^ satisfies the hypothesis.
                                
                                
                                First determine (b_j)_j=0^ so that v^*=∑_j=0^b_jφ_j(x) is 
                                an _1-minimizer of (<ref>), 
                                v^* ∈_, s(1/m(s, ) Λ_s, Λ_s) with m=1.
                                Then, by (<ref>), 
                                
                                    χ_M(, - 2)u-∑_j=0^(a_j-b_j)φ_j
                                    _L^2(M)< _0.
                                
                                Take d_j=a_j-b_j. Then 
                                v(x)= ∑_j=0^ d_j φ_j(x) satisfies (<ref>).
                                
                                
                                  §.§.§ Minimisation with finite interior spectral  data with errors
                                
                                In this section we consider an approximate construction  when there 
                                is a δ-error
                                 in FISD. 
                                  We assume that we are given the ball (B_e(r_0), g^a) and the pairs
                                 (λ^a_j, φ^a_j|_B_e(r_0)) with j=0,1,2,…,J.
                                
                                We assume
                                that these data are δ-close to
                                ISD of some manifold (M,g,p)∈_ n
                                in the sense of
                                Definition <ref>
                                with intervals I_p⊂,  p=0,1, …, P covering the spectrum of -Δ_g in 
                                [0, δ^-1+ δ].
                                We will use parameters ,∈_+ and P_0∈_+ satisfying 
                                (<ref>)-(<ref>), (<ref>), and  (<ref>).
                                Note that then  ≤≤ J and that
                                below we will use  (λ^a_j, φ^a_j|_B_e(r_0)) with j=0,1,2,…,.
                                Denote  𝒥_p={j∈_+; λ^a_j∈ I_p} and n_p is the number of elements in
                                𝒥_p.
                                
                                Then, for 
                                any p there is A^p∈ O(n_p), p=1,2,…,P_0 such that, if
                                j ∈𝒥_p then 
                                φ_j= ∑_k ∈𝒥_p A^p_jkφ_k
                                
                                satisfies φ_j-φ_j^a_L^2(M)<δ, where φ_k are the eigenfunctions of Δ_g.
                                Note that ∑_p=0^P_0 n_p =+1.
                                 We use below the matrix E∈ O(+1),
                                
                                
                                E=[e_jk ]_j, k=0^,    
                                e_jk=⟨|φ_k, φ_j_L^2(M)
                                and note that e_jk=0  if _j, _k do not lie in the same I_p.
                                
                                Let b =(b_0, b_1, …, b_) ∈^+1 then, for
                                b^a=E( b), b^a =(b_0^a, b_1^a, …, b^a_) we have
                                ∑_j=0^   b_j^a φ_j(x)=∑_j=0^ b_j  φ_j(x).
                                
                                
                                
                                Also, let ω_j  be the center point of the interval I_p containing
                                 λ_j^a so that |λ_j^a-ω_j|<δ.
                                
                                
                                
                                
                                
                                 The main goal of this section is to prove
                                 
                                 Let 
                                _0,_1,_2  satisfy (<ref>).
                                
                                
                                
                                
                                 Let    satisfy 
                                (<ref>), 
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                 satisfies (<ref>) and  satisfy (<ref>),
                                
                                δ satisfies (<ref>),
                                 and let J satisfies
                                
                                
                                  
                                  
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                J_0(δ)≤ J ≤ 2^n/2^n   J_0(δ) ,   where J_0(δ)=(2δ)^-n/2.
                                
                                Then  the following is valid:
                                
                                
                                
                                Let
                                 z_1,…,z_ L-1+N_1(σ)∈ B(p, r_0/4) be a σ-net.
                                Assume that g^a|_B_e(r_0) and ((λ_j^a,φ_j^a |_B_e( r_0)))_j=0^
                                is
                                 δ-close to  FISD g|_B(p,r_0)  and
                                 ((λ_j,φ_j |_B(p,r_0)))_j=0^ of  a manifold
                                  (M,g,p)∈_ n. 
                                 Also, assume that ã= (ã_j)_j=0^ satisfies
                                ∑_j=0^⟨|_j^a ^s  |ã_j|^2 ≤Λ_s^2,
                                and 
                                
                                    ũ^a(x)= ^*(ã)=∑_j=0^ã_j  φ_j(x),   for x∈ M.
                                
                                Let   ∈𝒜^(i).
                                
                                  Assume that we are given 
                                
                                g^a|_B_e(r_0),   ((λ_j^a,φ_j^a |_B_e( r_0)))_j=0^,    and   (ã_j)_j=0^.
                                 Let i ∈{L, …,   L-1+N_1(σ)} and
                                ∈𝒜^(i).  The data (<ref>) determine the set 
                                 ^a,*_2 and the function Ł_ a:^a,*_2→, 
                                
                                defined in (<ref>) and (<ref>),
                                
                                for which
                                the minimizer
                                
                                of Ł_ a in  ^a,*_2 is a sequence
                                    d^a=  d^a(,i)= ( d^a_j(,i))_j=0^∈^
                                    ,
                                 
                                
                                such that
                                v^a(x)= ^*(  d^a)=∑_j=0^d_j^a  φ_j(x), x∈ M,
                                satisfies, cf. (<ref>),
                                v^a ∈^̋s_ 2(s, ) Λ_s(M) ∩^̋0_ 2Λ_s(M),   v^a-χ_M(, -2)ũ^a_L^2(M)< _0.
                                
                                
                                  §.§.§ Proof of Theorem <ref>
                                The rest of this section will be devoted to the proof of Theorem 
                                <ref>. 
                                Similar to (<ref>), 
                                we introduce 
                                    ^*( b^a)= ∑_j=0^   b_j^a  
                                     φ_j(x),
                                       x ∈ M;
                                
                                and
                                 wave-type functions
                                      w^a(x, t)=^a(  b^a):=
                                   ∑_j=0^ b^a_j cos(√(λ^a_j) t) φ^a_j(x),    x ∈ B(p, r_0);
                                
                                
                                
                                      w(x, t)=( b^a)(x, t)=
                                W(^*( b^a))(x, t),   x ∈ M;
                                
                                
                                
                                      w(x, t)=
                                ∑_j=1^  b_j^a cos(√(ω_j) t ) φ_j(x),    x ∈ M;
                                
                                
                                
                                      w^a(x, t) =^a( b^a):= 
                                
                                ∑_j=1^  b_j^a cos(√(_j^a) t ) φ_j(x),    x ∈ M,
                                
                                where we recall that W is defined by W(v)=w where w satisfies (<ref>),
                                and
                                      ^a_ , s(a_1, a_2) ={  b^a ∈^( +1):   ∑_j=0^⟨|_j^a ^s |   b_j^a|^2
                                ≤   a_1^2,  ∑_j=0^| b^a_j|^2 ≤ a_2^2}
                                      ^a_, s(_*;a_1, a_2, α)=
                                {  b^a ∈^a_ , s(a_1, a_2) ;  ^a(b^a) _L^2(Γ̃(z_ℓ, _ℓ))≤_*, ℓ∈ K_i }.
                                
                                We note that (see (<ref>) and (<ref>))
                                ^*( b^a)= ^*(E^-1 b^a),  ( b^a)=  (E^-1 b^a).
                                
                                 Let 
                                 b^a ∈^a_ , s(  (, s) Λ_s, Λ_s).
                                
                                  
                                  If δ <1 satisfies  (<ref>) 
                                
                                then, 
                                      w-w^a_L^2(B(p, r_0)× (-2D, 2D))≤1/ 4_2 .
                                
                                 
                                
                                
                                 
                                 Proof.
                                 Due 
                                 to (<ref>) and (<ref>), 
                                 for j, k ∈𝒥_p,
                                |√(λ^a_j)-√(ω_k)| ≤ 2 √()δ,  φ_j-φ_j^a_L^2(B(p, r_0))≤δ,  φ_j^a_L^2(B(p, r_0))≤ 2.
                                
                                Using this, we obtain for |t|≤ 2D the following estimates. First, the Schwartz inequality implies that
                                
                                
                                
                                
                                
                                
                                 w^a(·, t)-  w^a(·, t)  _L^2( B(p,r_0))   ≤   
                                (∑_j=0^ |b^a_j|)δ
                                   ≤   
                                 (+1)^1/2( ∑_j=0^ (b^a_j)^2)^1/2δ≤ 2 Λ_s ()^1/2δ.
                                
                                Also,
                                we see that 
                                  w^a(·, t)- w̃(·, t)  ^2_L^2( B(p,r_0))   ≤   ∑_j=0^ (cos (√(_j^a) t )-cos (√(ω_j) t ))^2 (b^a_j)^2
                                   ≤      (2D)^2 δ^2
                                 Λ_s^2 =  4D^2 Λ_s^2δ^2
                                  .
                                We have
                                w(x, t)=
                                 ∑_p=0^P  ∑_j,k ∈𝒥_p b_j^acos(√(ω_k) t ) A^p_jkφ_k(x)
                                = 
                                ∑_k=0^ (∑_j ∈𝒥_p A^p_jk b_j^a) cos(√(ω_k) t ) φ_k(x)
                                
                                and
                                
                                 w(x, t)=
                                ∑_p=0^P  ∑_j,k ∈𝒥_p b_j^acos(√(_k) t ) A^p_jkφ_k(x)
                                = 
                                ∑_k=0^ (∑_j ∈𝒥_p A^p_jk b_j^a) cos(√(_k) t ) φ_k(x),
                                
                                and as A^p are orthogonal matrices and |√(_k)-√(ω_k)| ≤ 2 ^1/2δ,  we see similarly to  (<ref>) and (<ref>)
                                  w(·, t)- w(·, t)  ^2_L^2( B(p,r_0))   ≤     4D^2 Λ_s^2δ^2
                                  .
                                Combining the above estimates with 
                                δ<δ_0(_2, , , Λ_s)=   1/ _2/Λ_s and =min(^-1 , (1+)^-1/2/100(1+D)^3/2L),
                                
                                
                                we obtain the claim.
                                
                                
                                By Definition <ref> we have 
                                
                                
                                      E _, s(
                                
                                1/2a_1, a_2  ) ⊂^a_, s( a_1, a_2  )
                                ⊂ E _, s(
                                
                                 2a_1, a_2  ).
                                
                                Note that the ℓ^2-norms of the sequences (b^a_j)_j=1^ do not depend on eigenvalues
                                and, therefore, the same holds for the exact and approximate data. Also, the ℓ^2-norms are invariant 
                                with respect to the operations involving orthogonal matrixes.
                                
                                
                                
                                Definitions of the sets of sequences in (<ref>) and (<ref>), Lemma <ref>
                                and formula (<ref>) imply that
                                
                                E_, s(_*-1/4_2;1/2 a_1, a_2, α)⊂^a_, s(_*;a_1, a_2, α)
                                ⊂  E_, s(_*+1/4_2;2a_1, a_2, α)
                                Let us use _*=1/2_2 and define
                                ^a,*_m=  
                                 ^a_, s(1/m_m, 1/2(, s) Λ_s, Λ_s, ),   m∈{1,2,4}
                                Using the notations in (<ref>), we see that
                                
                                E ^*_ 4⊂^a,*_2⊂ E^*_ 1.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                Consider the
                                quadratic function Ł_ a:^+1→,
                                Ł_ a( c)=
                                 c- a^2_^(+1),  Ł_E a( c)=
                                 c-E a^2_^(+1).
                                
                                
                                
                                
                                
                                cf. (<ref>). Note that Ł_ a( c)=Ł_E a(E c).
                                
                                
                                
                                
                                
                                
                                Let b^*∈^*_4 and
                                b^a,*∈^a,*_2 be minimizers of Ł_a and  Ł_Ea, respectively, that is
                                      Ł_a( b^*)=min_ b∈^*_4Ł_a( b) =:J_min(4),
                                and
                                      Ł_Ea( b^a,*)=min_ b^a∈^a,*_2 Ł_Ea( b^a)=:  J_min^a(2).
                                
                                Note that  we do not anymore consider _1-minimizers, but the minimizers.
                                Since
                                ^* and ^a,* are bounded and closed set in
                                ^ +1 such  minimizers exist.
                                When _1<  Λ_s/8,
                                Lemma
                                <ref> (iv) implies 
                                
                                |J_min(4)-J_min(1)|≤ 2(2 Λ_s  _1 +4 _1^2)<5Λ_s  _1 
                                 Using  (<ref>), (<ref>),
                                 and the fact that E is an isometry, we see that
                                
                                    J_min(1)≤ J_min^a(2)≤ J_min(4),  and   J_min^a(2)
                                    ≤ J_min(1)+5Λ_s  _1.
                                
                                These implies that  the minimizer b^a,* of function Ł_Ea in the set  ^a,*_2 satisfies b^a,*∈^a,*_2⊂ E ^*_ 1 and so we have that b̃^*=E^-1 b^a,*
                                is an _1-minimizer of Ł_a in the class ^*_ 1. 
                                We denote b̃^*=( b̃^*_j )_j=1^.
                                Let a =E^-1ã so that
                                
                                    ^*(a)=∑_j=0^  a_j  φ_j(x)=u(x).
                                
                                Then, by applying Lemma <ref> we see that  v^*= ∑_j=0^b̃^*_j φ_j  satisfies (<ref>).
                                Then, choosing d_j^a=ã_j-b^a,*_j,   j=0, 1, …,, we see that ṽ^a=∑_j=0^ d_j^a φ_j satisfies (<ref>).
                                This proves Theorem <ref>.
                                 Similarly to Remark <ref> and Theorem <ref>,
                                we see that if
                                the collection of
                                 g^a|_B_e(r_0) and ((λ_j^a,φ_j^a |_B_e( r_0)))_j=0^J
                                 is
                                  δ-close to  FISD of  a manifold
                                  (M,g,p)∈_ n then without loss of generality, we can assume that J satisfies (<ref>). Indeed, the eigenvalues λ_j  with index   j>  2^n/2^n   J_0(δ)
                                 are not used in the proof of Theorem <ref>.
                                
                                § CONSTRUCTION OF THE APPROXIMATE INTERIOR DISTANCE MAPS.
                                
                                
                                 §.§ Volume estimates
                                
                                
                                Our next goal is to approximately evaluate
                                the volume of M(α), see (<ref>) with b=0. 
                                
                                
                                There are uniform constants _0^* >0, (s)>1, depending only on  s, n, R, D, i_0 and r_0,
                                
                                 such that the following holds:
                                
                                Let _0 ≤_0^*.  
                                  Let  _1, _2 be defined by (<ref>) while 
                                  _0, be defined by (<ref>)
                                and (<ref>)–(<ref>). 
                                Assume that   we are given(g^a|_B_e(r_0); {(_j^a, φ^a_j|_B_e(r_0)) }_j=0^J) 
                                that is
                                 δ-close to FISD of
                                (M,g,p)∈_ n.
                                Here J satisfies by (<ref>). 
                                
                                Let also i ∈{ L, …,   L-1+N_1}, 
                                where N_1=N_1(σ) is defined  as in
                                Lemma 
                                <ref> (ii).  Assume that 
                                σ≤τ_0/2
                                  
                                where τ_0 is defined in Proposition
                                <ref> and let ∈𝒜^(i)
                                satisfy (<ref>).
                                
                                  Then
                                we can compute an approximate volume, ^a ((α)), of    the set M() that satisfies
                                |^a ((α))-((α))|≤(s) _0.
                                 Proof. Recall that
                                φ_0(x)=(M)^-1/2,   ℱ(φ_0)=(1, 0, 0, …), 
                                  φ_0_H^s =1   for  
                                s >0.
                                
                                The  interval I_0=(a_0, b_0) in Definition <ref> contains only
                                 _0=0. Thus φ_0^a|_B_e(r_0)
                                is a δ-close to φ_0|_B_e(r_0)=φ̃_0|_B_e(r_0). 
                                These allow us to evaluate ^a(M) so that |^a(M) -(M)|< C δ.
                                Using Theorem <ref>
                                we evaluate the 
                                 Fourier coefficients ( d^a_j)_j=0^
                                of v^a(x) which satisfies (<ref>) with
                                ũ= φ_0.
                                Let
                                ^a( M())=^a(M)  (∑_j=0^ ( d^a_j)^2)^1/2
                                Then, by (<ref>),
                                
                                |  ^a( M()) -((, -2 ))   | ≤ C(_0+δ).
                                
                                Since |( M())- ( M(,-2)) |<  L (cf. Lemma <ref>), 
                                (<ref>) implies
                                estimate (<ref>),  if _0 ≤_0^* with some uniform constants (s) and _0^*.
                                Here _0^* is defined so that δ <_0,   <_0 for 
                                _0 <_0^*, see (<ref>), (<ref>).
                                
                                
                                
                                
                                Next we  use FISD with errors to approximately find the distances from various
                                points x ∈ M to points z ∈ B(p, r_0/4). The  main tool is to
                                approximately
                                find the volumes of subdomains of M obtained by 
                                the slicing procedure. 
                                
                                
                                
                                
                                
                                
                                
                                For i ∈{L, …,   L-1+N_1(σ)} and β∈𝒜^(i),
                                
                                 M(β) are the domains defined in (<ref>) with  replaced by β.
                                We consider the intersection of slices,
                                ^*_ (i)(β) = ⋂_ℓ∈ K_i
                                 (B(z_ℓ, β_ℓ+2 σ)∖ B(z_ℓ, β_ℓ-2 σ))
                                
                                
                                
                                
                                
                                =(⋂_ℓ∈ K_i
                                  B(z_ℓ, β_ℓ+2 σ))∩(⋃_ℓ∈ K_i
                                 B(z_ℓ, β_ℓ-2 σ) )^c.
                                
                                Here for Ω⊂ M, Ω^c =M ∖Ω. 
                                
                                
                                
                                
                                
                                
                                
                                Note that
                                      ( (⋂_ℓ∈ K_iΩ_ℓ) ⋂Ω^c )=
                                ∑_ℓ∈ K_i(Ω_ℓ∪Ω) 
                                       - ∑_ℓ≠ℓ'=1^n (Ω_ℓ∪Ω_ℓ'∪Ω)
                                + … +(-1)^(L+1)( (⋃_ℓ∈ K_iΩ_ℓ ) ∪Ω)
                                - (Ω).
                                
                                By (<ref>),  M^*_ (i)(β) has form (<ref>)
                                with 
                                    Ω_ℓ= B(z_ℓ, β_ℓ+2 σ),  Ω=
                                     ⋃_ℓ∈ K_i B(z_ℓ, β_ℓ-2 σ).
                                
                                For any  _1,_2∈𝒜^(i) we have
                                (α_1) ∪(_2) = (_m),   where   (_m)_ℓ = max ((α_1)_ℓ, (_2)_ℓ).
                                
                                Therefore all terms in (<ref>) are of form (M()) for some 
                                ∈𝒜^(i). Thus,
                                using Lemma <ref>, we can approximately compute each term
                                of (<ref>) with error _0. Since there are 2^L+1 terms in (<ref>),
                                we
                                obtain the following result.
                                 Under the conditions of Lemma <ref>,
                                there exists _4(n, R, D, i_0, r_0) >0 and ∈ (0,1),
                                
                                 depending only on n, R, D, i_0 and r_0,
                                
                                 with the following property:
                                
                                Let  0<_4 <_4(n, R, D, i_0, r_0).   
                                It is possible to evaluate approximate volumes,
                                ^a(M^*_ (i)(β)), of the sets
                                M^*_ (i)(β) of form (<ref>). Moreover, 
                                | ^a(M^*_ (i)(β))- (M^*_ (i)(β)) | ≤_4,
                                  if  _0 ≤_4.
                                
                                
                                 §.§ Distance functions approximation
                                
                                A function r(·)∈ C^0, 1(B(p, r_0/4)) is an  interior distance function  
                                if there is x∈ M such that
                                r(z) =r_x(z) = d(x,z), for any z ∈ B(p, r_0/4).
                                
                                The interior distance functions determine the interior distance map
                                
                                R_M:  (M, g) → L^∞(B(p, r_0/4)),   
                                R_M(x) = r_x(·).
                                
                                The map R_M or, more precisely, its image
                                
                                R_M(M):= { r_x(·),   x ∈ M }⊂ L^∞(B(p, r_0/4)),
                                
                                may be used to reconstruct (,g).  Namely, in   <cit.>,
                                <cit.>
                                it was shown  how to reconstruct
                                (N,  g|_N), where
                                N=M ∖ B(p, r_0/50),
                                from the knowledge of  boundary distance functions
                                R_N(N)
                                = {r^N_x(·) ∈ L^∞( N);  x∈ N}, 
                                  
                                r_x^ N(z)=d_N(x,z),
                                
                                where d_N is the distance in N.
                                Later, in Section <ref> we show that 
                                a Hausdorff approximation R^*_M to R_M(M) makes it possible to
                                construct an approximation R^*_N to R_ N(N).
                                
                                
                                Thus, our next goal is
                                to construct a desired approximation R_M^*.
                                 To this end, we use the volume approximations
                                of the previous subsection.
                                
                                First, for z,z'∈ B(p,r_0/2), we define  an approximate distance d^a(z, z')
                                using the metric g^a.
                                Then Definition <ref> (iv) together  with convexity of B(p, r_0), 
                                see (<ref>),
                                imply that
                                
                                |d^a(z, z') -d(z,z')| ≤σ,   if  δ < σ.
                                 Recall that above we have used a parameter σ>0 
                                
                                 which  satisfy σ≤τ_0/2 
                                and we have chosen points {z_1,…,z_ L-1+N_1(σ)}⊂ B(p, r_0/4)
                                such that {z_1,…,z_L-1} is a τ_0-net in B(p, r_0/4). Moreover, the set {z_L,…,z_ L-1+N_1(σ)}
                                is a maximal σ-separated set in
                                B(p, r_0/4), see (<ref>).
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                For any i∈{ L, …,   L-1+N_1(σ)}
                                 and β=(β_ℓ)_ℓ=1^ L-1+N_1(σ)∈^ L-1+N_1(σ),r_0/8 < β_ℓ < 2D,
                                 cf. (<ref>), we define 
                                 ^(i)(β)=β^(i)∈^ L-1+N_1(σ), where
                                       β^(i)_ℓ=β_ℓ, if ℓ∈ K_i,
                                      β^(i)_ℓ=0, if ℓ∉K_i.
                                 Then, ^(i)(β) ∈𝒜^(i).
                                
                                Observe that, for any x ∈ M ∖ B(p, 3r_0/8+σ) and ℓ=1, …, L-1+N_1(σ)
                                
                                there is
                                β_ℓ(x)∈σ_+
                                such that
                                 β_ℓ(x)-σ≤ d_M(x,z_ℓ)≤β_ℓ(x)+σ. Therefore,
                                B(x, σ) ⊂ B(z_ℓ, β_ℓ(x) +2 σ) ∖ B(z_ℓ, β_ℓ(x)-2σ),
                                so that, due to (<ref>),
                                
                                    (B(z_ℓ, β_ℓ(x) +2 σ) ∖ B(z_ℓ, β_ℓ(x)-2σ) 
                                    ≥1/σ ^n.
                                
                                Taking into account this inequality together with (<ref>) we require
                                _4  ≤1/4σ ^n.
                                
                                Thus, for i∈{ L, …,  L-1+N_1(σ)},   the 
                                 volume and the approximate volume
                                of the set
                                 M^*_ (i)(β^(i)(x)), β^(i)(x)=^(i)(β(x)) satisfy
                                    (M^*_ (i)(β^(i)(x))) ≥ 4 _4,   ^a(M^*_ (i)(β^(i)(x))) ≥ 3 _4,
                                
                                
                                where we use (<ref>).
                                 The above considerations motivate the following definition. 
                                In order to use only finitely many indexes β, in the following we are going to 
                                 consider  β=(β_i)_ℓ=1^ L-1+N_1(σ) where β_i ∈σℤ_+, β_i≤ 2D.
                                Let β =(β_ℓ)_ℓ=1^ L-1+N_1(σ)∈σℤ_+^ L-1+N_1(σ)⊂ℝ_+^ L-1+N_1(σ). Such sequence
                                 β is called  admissible, if  r_0/8≤β_ℓ≤ 2D and 
                                 for all indexes
                                 i∈{L,…,  L-1+N_1(σ)},
                                 the modified index
                                  β^(i)=^(i) (β)∈𝒜^(i) satisfies
                                ^a(M^*_(i)(β^(i))) ≥ 3_4.
                                
                                We define the set ℬ = 
                                {β∈σℤ_+^ L-1+N_1(σ); β}.
                                
                                For any x ∈ M∖ B(p, 3r_0/8+σ), there exists an admissible
                                β∈σℤ_+^ L-1+N_1(σ)
                                such that
                                |d(x, z_ℓ)-β_ℓ| ≤ 2 σ, for ℓ∈{1,2,…,  L-1+N_1(σ)}.
                                
                                Conversely, there is >0 depending only on n, R, D, i_0 and r_0,
                                
                                 such that, if β is admissible, then
                                 there is x=x_β∈ M ∖ B(p, 3r_0/8- σ) such that,
                                for all
                                ℓ∈{1,2,…,  L-1+N_1(σ)}, we have
                                 |β_ℓ-d(x,z_ℓ)|≤σ.
                                 Proof.
                                 The first statement follows from considerations before Definition <ref>.
                                
                                On the other hand, assume that β∈ℬ. Then
                                equations (<ref>) and (<ref>) guarantee  that, for any i∈{L,…,  L-1+N_1(σ)},
                                 there is x_i   ∈ M^*_ (i)(𝒯^(i)(β)).Moreover, we have 
                                |d(x_i, z_ℓ) -β_ℓ| ≤ 2 σ,  for  ℓ∈{1, …, L-1}∪{ i}.
                                Moreover, in view of (<ref>), for j, k ∈{L, …,   L-1+N_1(σ)},
                                
                                d_M(x_j, x_k) ≤ |H^L(x_j)-H^L(x_k)| ≤ 4 √(L)σ.
                                 Defining
                                = 4 √(L)+3,
                                
                                and taking x=x_ i_1  with  arbitrary i_1, we see that
                                x ∈  M ∖ B(p, 3r_0/8-σ) and that  (<ref>)
                                is satisfied. For the points {z_ℓ: ℓ∈{1,…,  L-1+N_1(σ)}⊂ B(p, r_0/4), let 
                                
                                    V_ℓ ={y∈ B(p, r_0/4): z_ℓ is the unique closest point to y in {
                                    z_ℓ'}} ,
                                
                                where ℓ=1, …, L-1+N_1(σ),  be the corresponding  Voronoi region.
                                With any  β∈ℬ we then associate
                                a piecewise constant function   r_β∈ L^∞(B(p, r_0/4)) by defining 
                                r_β(z)= β_ℓ, for z ∈ V_ℓ. Clearly,
                                
                                d_L^∞(M)(r_β,  r_x) ≤σ,   =+2.
                                
                                Let
                                
                                    R^*_M, >={r_β(·): β∈ℬ }⊂  L^∞(B(p, r_0/4)).
                                
                                 Choose a  maximalσ-net {x_1, …, x_ N_2(σ)}⊂ B(p, r_0/2)
                                by adding to z_1, …, z_ N_0(σ) a σ-net 
                                z_ N_0(σ)+1, …, z_N_2(σ)   in B(p,r_0/2) ∖ B(p, r_0/4).
                                 Again, using Lemma <ref>,
                                we see that  N_2(σ) ≤σ^-n.
                                Next we define
                                
                                r_k(z)=d^a(x_k, z_ℓ),   for z ∈ V_ℓ,    k=1, …,  N_2(σ),  ℓ=1,…,   L-1+N_1(σ);
                                      R^*_M,<={r_k(·): k=1, …,  N_2(σ)}⊂ L^∞(B(p, r_0/4)), 
                                       R^*_M =R^*_M,> ∪ R^*_M, <.
                                 In Figure <ref>, we consider the set
                                
                                X={x_β : β∈ℬ}∪{x_1, …, x_ N_2(σ)}⊂ M.
                                
                                Thus, denoting = 2+2+1, see (<ref>) and  (<ref>),
                                 we obtain
                                
                                We have
                                
                                d_H(R_M(M), R^*_M) ≤σ,
                                
                                where d_H is the Hausdorff distance in L^∞(B(p, r_0/4)).
                                
                                
                                §  PROOF OF THEOREM <REF> AND PROPOSITION <REF>
                                
                                
                                 §.§ From interior distance functions to boundary distance functions
                                
                                
                                
                                By standard estimates for the differential
                                of the exponential map, see <cit.> the diameter of the sphere B(p,r),   r<r_0, is bounded
                                
                                ( B(p,r))≤π r sinh( √(K) r)/√(K) r≤π r  cosh(π/2)≤ 10  r,
                                
                                where we use condition 
                                  (<ref>). Let N=M∖ B(p,r_0/25).
                                
                                
                                Let  x ∈ M∖ B(p,r_0/4)
                                and y∈ N and z∈ B(p,r_0/4), let
                                      f(y, x, z)=d_N(y,z)+d_M(z,x),
                                      f(y,x)=min_z_1∈ B(p,r_0/4)f(x, y; z_1),
                                where d_N  and d_M are the distances in N and M, respectively.
                                Then,
                                
                                d_N(y,x)=f(y,x)
                                
                                
                                 Proof.
                                Clearly, as d_M(z,x)≤ d_N(z,x) and  a  shortest curve
                                in N from y to x intersects the sphere B(p,r_0/4), we see that  d_N(y,x)≥ f(y,x).
                                
                                On the other hand let z'= argmin_z(f(y, x; z)) and μ([0, f(y,x)] be the
                                corresponding union of the distance minimizing paths from y  to z' and from z' to x for
                                which the minimum in (<ref>) is achieved.
                                Denote s_1= d_N(y,z') and consider μ([s_1, f(y, x)]. We  show next that
                                μ([s_1, f(y, x)] ⊂ N.
                                 If this is not the case, there would exists s_1 < s_2 <s_3 <f(y,x)
                                such that μ(s_1), μ(s_3) ∈ B(p,  r_0/4), μ(s_2) ∈ B(r_0/25)
                                and μ[s_3, f(y,x)] ⊂ M ∖ B(p,r_0/4). Then,
                                
                                s_1 ≥ r_0(1/4-1/25),     s_2 -s_1 ≥  r_0(1/4-1/25),
                                   s_3 -s_2 ≥  r_0(1/4-1/25).
                                
                                On the other hand, consider a curve μ'([0, l]) which is parametrised by the arclength
                                and consists of the radial path from μ(s_3)
                                to y' ∈ B(r_0/25) followed by a shortest
                                path along B(r_0/25) from y' to y. Due to (<ref>) and (<ref>),
                                
                                l ≤ r_0(10/25+1/4-1/25) <3 r_0 (1/4-1/25) ≤ s_3.
                                
                                Taking the union of the path μ'([0, l]), connecting
                                 μ(s_3)
                                to y', and the path
                                 μ(s_3, f(y,   x)), connecting y' to x, we get a contradiction to definition
                                (<ref>). Thus, μ([s_1, f(y, x)]) ⊂ N, i.e.,  d_N(y, x) ≤ f(y, x).
                                
                                
                                Next, using the already constructed set R^*, see (<ref>)
                                together with Lemmata <ref> and <ref>,
                                 we construct a set R^*(N) ⊂
                                L^∞( N)  which  approximates 
                                 
                                
                                R^ N(N)= {r_x^ N∈ L^∞( N):  x∈ N},
                                
                                 where
                                r_x^ N(z)=d_N(x,z),
                                 for  z∈ B(p, r_0/25).
                                Let R^*  be the set given in (<ref>),
                                which satisfies (<ref>) be given. Then it defines a set R^*(N) ⊂ L^∞( N)
                                such that
                                
                                d_H(R^ N(N),  R^*(N)) ≤σ,   = 2+2+1 .
                                
                                Here  is defined in (<ref>) and  is defined in (<ref>).
                                
                                Note that here we assume that δ satisfies (<ref>), σ satisfies (<ref>) with the
                                related equations for _4, _0, etc.
                                 Proof. The proof is based on the construction of R^*(N) which satisfies (<ref>).
                                
                                Observe first that it follows from the proof of Lemma <ref> that,
                                if x, y ∈ B(p, r_0/4) ∖ B(p, r_0/25) ⊂ N,
                                then
                                
                                d_N(x, y) ≤r_0/2 + 8 r_0/25,
                                
                                so that a shortest path in N connecting x and y lies in B(p,  r_0). Thus it is possible, using (<ref>),
                                to construct an approximation r̃_x^ N: N→ that satisfies
                                r_x^ N-r̃_x^ N_L^∞( N)≤σ,
                                
                                with a uniform constant , cf. (<ref>). Denote
                                R^*_<(N)={r̃_x^ N;  x ∈ B(p, r_0/4) ∖ B(p, r_0/25) }, then
                                
                                d_H(R^ N(B(p, r_0/4)),  R^*_<(N)) ≤σ,
                                
                                for δ< δ_0,
                                cf. construction of R^*_< in subsection <ref>.
                                
                                
                                Next, let
                                
                                R^*_c={r ∈ R^*: min_z ∈ N(r(z)) ≥r_0/8}
                                For y,z ∈ B(p, r_0/4) ∖ B(p, r_0/25) denote by d^a_N(y,z) the distance between y and z in the metric g^a along the curves lying in B(p, r_0/2) ∖ B(p, r_0/25).
                                For each r ∈ R^*_c we define
                                      r̃^ N∈ L^∞( N):   r̃^ N(y)=
                                inf_z ∈ B(p, r_0/4)(d^a_N(z, y)+ r(z)  );
                                
                                       R^*_>(N)={r̃^ N(·):  r ∈ R^*_c  }.
                                
                                Then,
                                  with R^*(N)= R^*_<(N) ∪ R^*_>(N), we have that
                                
                                d_H(R^ N(N),  R^*(N)) ≤ (2+)σ=σ.
                                
                                Here σ error comes from an approximation of d_N(y, z),
                                see
                                 (<ref>), and
                                2σ error comes from approximating d_M(z, x) and d_N(y, z) in formula (<ref>), see also (<ref>)-(<ref>) and (<ref>). At last,
                                we use again that δ satisfies the  uniformly bound  (<ref>).
                                
                                
                                
                                Recall that the metric tensor g  on B(p,r_0) is a representation of a
                                metric in Riemannian normal coordinates and the C^2,α-norm of the metric
                                is uniformly bounded. Using the fundamental equations of the Riemannian geometry, <cit.>,
                                we have that the shape operator S of the surface B(p,r),   r <r_0, can be given in the Riemannian
                                normal coordinates centered at p
                                in terms
                                of the metric tensor
                                as S=g^-1_ν g, where ν  is the unit normal vector of  B(p,r). Taking r=r_0/25, we see
                                 that the  C^1,α-norm of the shape operator S of N is uniformly bounded.
                                Also, by  
                                  (<ref>)
                                 
                                the boundary injectivity radius of (N,g|_N) is bounded below by 24/25 i_0. As the sectional curvature
                                of M and the second fundamental form (that is equivalent to the shape operator) of its submanifold N
                                are bounded, the Gauss-Codazzi equations imply that the sectional curvature of N is bounded.
                                As the metric tensor of M is bounded in normal coordinates in B(p,r_0), we see that
                                the (n-1)-dimensional volume of  N= B(p,r_0/25) is bounded from below by a uniform constant.
                                Thus by Cheeger's theorem, see <cit.>,
                                the injectivity radius of N is bounded from below by a uniform constant.
                                
                                Summarising the above, the Ricci curvature of (N,g|_N  )   is uniformly bounded in C^α, the
                                second fundamental form of N is uniformly bounded in C^1,α,
                                and the diameter and injectivity radii of N and N, and the boundary injectivity radius of (N,    N) are
                                uniformly bounded.
                                By <cit.>, using the knowledge of the set, R^*(N) of approximate boundary distance functions,
                                which are σ-Hausdorff close
                                to  the set, R^ N(N)  of the boundary distance functions  of manifold (N,g|_N), one can construct
                                on the set  R^*(N)  a new distance function
                                d^*_N:  R^*(N) × R^*(N) →_+, such that
                                
                                d_GH((N,d_N),(R^*(N),d^*_N))≤( σ)^1/36,
                                
                                with a uniform >0.
                                
                                
                                Having constructed (R^*(N),d^*_N) we can now construct an approximate metric space (M^*, d^*_M)
                                which is ( σ)^1/36- close to (M, d_M).
                                Indeed, let x, y ∈ N and μ[0, l],   l=d_M(x, y)
                                be a shortest between x and y. If  μ[0, l] ⊂ N then d_M(x, y))=d_N(x, y). If, however,
                                μ[0, l] intersects with B(p, r_0/25)  then, due to the convexity of B(p, r_0/25),
                                there are 0<s_1 <s_2<l
                                such that
                                μ[0, s_1] ⊂ N,   μ[s_1, s_2] ⊂ B(p, r_0/25),   μ[s_2, l] ⊂ N.
                                
                                Therefore,  similar to Lemma <ref>, we obtain
                                
                                Let x, y ∈ N. Then
                                      d_M(x, y) =
                                
                                      min(d_N(x, y), min_z_1, z_2 ∈ B(p, r_0/25)[d_N(x, z_1)+
                                d_M(z_1, z_2)+d_N(z_2, y)  ]  ).
                                
                                
                                Next define, for r̃^ N_1, r̃_2^ N∈ R^*(N),
                                      d^*_M(r̃_1^ N, r̃_2^  N)= 
                                      min(d^*_N(r̃_1^ N, r̃_2^ N), min_z_1, z_2 ∈ B(p, r_0/25)[r̃_1^ N(z_1)+ d^a(z_1, z_2)+r̃_2^ N(z_2)  ]  )
                                
                                Using (<ref>) together with (<ref>),
                                (<ref>) and
                                 (<ref>),
                                we see that 
                                d_GH((N, d_M),  (R^*(N), d^*_M) ≤  (2 +1)( σ)^1/36 if  σ≤ (σ)^1/36.
                                
                                Here (N, d_M) is  the manifold N with the distance function inherited from M and
                                 δ <δ_0, cf. (<ref>).
                                
                                
                                 Let us define the disjoint union
                                M^*= R^*(N) ∪ B(p, r_0/25). Next we  define a metric d^*_M on this set.
                                To this end, consider first r̃^ N∈ R^*(N),  y ∈ B(p, r_0/25). Recall,
                                see the proof of Lemma <ref>,
                                  that the set R^*(N) is bijective with R^*_c ∪(B(p, r_0/4) ∖ B(p, r_0/25)).
                                In the case when r̃^ N is obtained from r ∈ R^*_c, we define d_M^*(r̃^ N, y)=r(y).
                                 Moreover, in the case when
                                 r̃^ N is obtained from x ∈ B(p, r_0/4) ∖ B(p, r_0/25), we define d_M^*(r̃^ N, y)=
                                 d^a(x, y).
                                At last, if x, y ∈ B(p, r_0/25), we take d_M^*(x, y) =d^a(x, y).
                                
                                 It follows from (<ref>) together with equations
                                (<ref>), (<ref>),
                                (<ref>) and considerations preceding Lemma <ref> that
                                
                                d_GH((M^*, d^*_M), (M, d_M)) ≤ (2 +1)( σ)^1/36.
                                
                                Summarizing, we obtain
                                 Let R^*  satisfy  (<ref>) and
                                M^*= R^*(N) ∪ B(p, r_0/25) with metric d^*_M. Then,
                                
                                
                                
                                d_GH((M, d_M), (M^*, d_M^*)) ≤σ^1/36,   = (2 +1) ^1/36.
                                
                                
                                 §.§ Proof of Theorem <ref> and Proposition <ref>
                                 Proof of Proposition <ref>.
                                
                                To prove the statement of the 
                                
                                  Proposition, 
                                we collect all the previous estimates. 
                                
                                The aim is to find the relation between the final error 
                                (i.e. d_GH((M, d_M), (M^*, d_M^*)) ≤)
                                and the initial error δ.
                                We proceed by following the chain of  relations:
                                ↦σ↦_4 ↦_0 ↦_1 ↦↦_2
                                ↦↦↦δ.
                                 To obtain inequality (<ref>) from
                                (<ref>) we set
                                
                                σ = ( /)^36
                                
                                 and use it in
                                
                                
                                
                                
                                (<ref>),
                                (<ref>), (<ref>) and (<ref>) with Λ_s=1
                                to determine values of _4, _0 and _1 by setting      _4 =^36n/4 ^36n, _0 = 
                                _4 ≤1/10,
                                
                                 and
                                      _1 = ^72n =^2 /160 ^2 ^72n.
                                 
                                To define  so that  (<ref>), (<ref>),
                                and (<ref>)  are valid, we set  
                                
                                
                                = _1^1/b(s),   = min(^-1/(2n)^-36, , r_0/32 ).
                                
                                Here we have used that σ = _1^1/(2n)^-1/(2n)^-36 and noticed that b(s) < 2n.
                                
                                 From (<ref>)  and  (<ref>)
                                 we get
                                 _2=
                                
                                  (_1^1/b(s))^s/(s-1)/( exp[ (   4L 
                                _1^-1  (_1^1/b(s))^-2+θ/2 exp(^-c_200_1^-c_200/b(s)) 
                                  
                                  )^1/β]  )^s/(s-1) with _1 given by (<ref>).
                                 
                                 Finally, to choose  and δ so that  (<ref>), (<ref>), (<ref>)(<ref>) 
                                 
                                 are satisfied,
                                 we set
                                   =( _2/8;  , 1)=
                                ^-n   8^-n/s_1^-n/b(s)_2^-n/s, 
                                with _2 given by (<ref>),
                                and choose δ so that
                                δ   ≤   2^-n/2^-n  ()^-1_2 
                                   =  
                                  
                                
                                
                                2^-n/2^-n^-1^n  8^ n/s_1^n/b(s)_2^1+n/s
                                   =    _1^exp[- _1^-exp(  _1^-) ],  
                                with
                                      
                                C_34= 1/b(s)(n+ s+n/s-1)
                                ,   
                                C_35= (s+n)/(s-1)( 4L C_12^-(2- θ/2))^1/β,
                                
                                      
                                C_36=^-c_200/β,   
                                C_37= 1/β(1+ 1/b(s)(2-θ/2)),   
                                 C_38=c_200/b(s),
                                
                                      
                                C_39= 2^-n/2^-n^-1    8^ n/s^(s+n)/(s-1)+n. 
                                We use the inequality x≤exp(x)  to bound from below the right hand side of the  estimate above to obtain,
                                by calling = max(C_34,C_37,C_38,1/(2n)),
                                δ≤exp[-exp((C_39^-1+ C_35 +  C_36 )_1^- ) ].
                                
                                Notice that (<ref>) is also satisfied,  by replacing  C_39 with =/(C_33^36^1/(2n)) and by including 1/(2n) in .
                                Assuming 0< δ≤exp(-e),
                                we get
                                ( ^-1+ C_35 +  C_36 )/ln(ln1/δ)≤_1^,
                                
                                Let τ_0 be the uniform constant introduced in Proposition <ref> and define
                                 
                                 C_44 = min(1000^- , ^(^36τ_0/2)^2n ).
                                 
                                In this way we can set in (<ref>) the two constraints (<ref>) and _1 ≤ 1/ 1000 (derived from (<ref>) with Λ_s=1)  and obtain
                                
                                
                                    δ≤δ^*, δ^* = min( exp(-e), exp[-exp [C_44^-1( ^-1+ C_35 +  C_36 )]]).
                                
                                Finally by using (<ref>) to rewrite _1 in (<ref>), and defining =1/(72n ) and
                                = ( ^-1+ C_35 +  C_36 )^ ^-1/(72n),
                                we obtain (<ref>).
                                
                                 Proof of Theorem <ref>.
                                Let δ≤δ^* and let the ISD of M^(i),   i=1,2 be δ-close. 
                                
                                
                                Take the finite collection
                                
                                𝒟 = ((B^e(r_0), g^(1)), {(_j^(1), φ_j^(1)) }_j=0^J), 
                                
                                where the index (1) is related to the IDS of M^(1).
                                
                                By construction the data 𝒟_0 are δ-close to the ISD of both M^(1) and
                                M^(2). By 
                                
                                 Proposition <ref> 
                                
                                the metric space (M^*, d^*_M) constructed with these data is
                                -close to both (M^(i), d^(i)), i=1,2,
                                where  is given by  the right hand side  of (<ref>).
                                We then conclude by triangular inequality,   
                                for any 0 <δ≤δ^*,
                                
                                d_GH((M^(1), d^(1)),   (M^(2), d^(2))) ≤ 2
                                We now extend this estimate to the case δ∈ (0, exp(-e)], when δ^* < exp(-e).
                                To this end, observe that the definition of the GH-topology and (<ref>) imply that:
                                d_GH((M^(1), d^(1)),   (M^(2), d^(2))) ≤ D,
                                for any δ.
                                By combining the latter inequality and (<ref>) 
                                we obtain the inequality (<ref>) with
                                =
                                
                                max(2,
                                D(ln(-lnδ^*))^).
                                  Acknowledgements RB and ML were partially supported by Academy of Finland, projects
                                 303754, 284715 and 263235.
                                YK was partially supported by
                                 EPSRC grant EP/L01937X and Institute Henri Poincare.
                                
                                 Table of constants C_k, c_k, τ_0 and s.
                                
                                
                                 Note that all constant depend on
                                n, R, D, i_0, r_0 and variables  in brackets.
                                
                                
                                
                                
                                Name      Introduced in / Notes    Name      Introduced in / Notes
                                
                                
                                        Thm. <ref>          Thm. <ref>
                                       Prop. <ref>           Cor. <ref>
                                        Cor. <ref>      (<ref>)
                                C^(har)   (<ref>)   C^(Lip)   (<ref>)
                                   (<ref>)       Prop. <ref>
                                     Prop. <ref>       Prop. <ref>
                                    Lemma  <ref>   (θ)    see (<ref>), we use θ = 1/2
                                   (<ref>), we use
                                 =1-θ/2   c_200   (<ref>)
                                c_205 ( θ)   (<ref>), Appendix    c_206(,θ)   (<ref>), Appendix
                                   (<ref>)   τ_0   Prop. <ref>
                                   Lemma <ref>   (s)   Lemma <ref>
                                b(n)    Lemma <ref>   (s)    Lemma <ref>
                                s   (<ref>)   (s)   (<ref>)
                                (s;)   Lemma <ref>      (<ref>)
                                   (<ref>)   (s)   Lemma <ref>
                                    Lemma <ref>      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)       Lemma <ref>
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      (<ref>)
                                   (<ref>)      
                                
                                
                                
                                § APPENDIX
                                
                                
                                 §.§  Calculation of c_206(,θ) in Theorem <ref>
                                 
                                To prove Theorem <ref> we need to show that the solution w of the wave equation
                                
                                P(y,D) w(y)= q̃(y),   y ∈Ω(T)⊂^n+1
                                can be estimated in the set (z,γ,T)⊂^n+1 in (<ref>) that is between two double cones of the spacetime,
                                i.e. Σ(z,γ,T) ⊂(z,γ,T)  ⊂Σ(z,0,T). Here, γ>0 is the
                                parameter that indicates how close the set (z,γ,T)  is to the optimal double cone Σ(z,0,T).
                                Theorem <ref> is proven by applying a proper iterative procedure and the dependency of the
                                coefficient c_206 on γ is crucial for our considerations.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                The calculation of  c_206 can be consider as the final step of a long geometric construction.
                                In order to understand it we summarize the previous steps with related references.
                                
                                In Section 3 of <cit.> we calculated the parameters of the inequality associated with a (conormally) pseudo-convex function ψ with respect to the wave operator P(y,D).
                                Then we used this property to calculate the coefficients of the Tataru inequality (recalled in the following section <ref>)
                                
                                    e^-ϵ |D_0|^2/2τ e^τ f u_1,τ≤ c_1,T τ^-1/2e^-ϵ |D_0|^2/2τ e^τ f P(y,D)u_0 + c_2,T e^-τ R_2^2/4ϵe^τ f u_1,τ
                                
                                 and to prove the local stability of the unique continuation for the wave operator. 
                                
                                In  <cit.> we used the previous result to prove the global stability of the unique continuation for the wave operator. As recalled in the following section <ref>, the proof is based on the iteration N times of the a local stability for the 'low temporal frequency' component of the solution u of the wave equation:
                                
                                    A(D_0/ω)b((y-y_j)/r)u_j_H^1≤ c_155,jexp(-c_132μ_j^α^2),  ∀ ω≤μ_j^α/(3c_131).
                                
                                Moreover in Section 3.1. of <cit.> and Appendix A of <cit.>  we applied the stability result in the domain of influence of a cylinder. 
                                
                                
                                In the case of the present paper, the mentioned domain of influence is called Σ(z,0,T) in Theorem 2 and, according to the iterative procedure, it contains a covering of the set Λ=(z,γ,T). The balls of the covering have radius 2R, that depends on the distance to the boundary,
                                on the regularity and pseudo-convexity property of the function ψ, and on extra constraints imposed by the Tataru inequality. The local stability step holds for smaller balls with radius r, and r < R. In Table 1 we summarize these values and in particular we obtain, up to a multiplicative constant,
                                
                                r∼γ^58.
                                
                                The ∼ symbol is defined precisely in subsection <ref>.  
                                By  construction and for (<ref>), one can calculate the number of local steps of the iteration (see also Table 2)
                                
                                N∼γ^-58(n+1).
                                
                                These two values combined with the calculation of the coefficients for the local and the global stability lead to the following relationship between c_206 and γ
                                    c_206  = ζ_1  exp(  γ^-ζ_2), ζ_1, ζ_2.
                                
                                We will prove that formula (<ref>) plays a big role for the calculation of ζ_2.
                                
                                In both articles <cit.>
                                we used consistent notations for the geometric quantities and labeled the important coefficient as c_h, with a unique h≥ 100 in order to be able to follow the construction of the final parameters. One can find them  in those papers by searching for the corresponding index h.
                                
                                Here in this Appendix our focus is the dependency of all the parameters (in particular c_206) on the quantity γ,
                                
                                since this reflects the cost of getting close to the cone of dependence. For this reason in the following section <ref> we quickly introduce the main relationship between γ and the used Gevrey function localizers, and in the next sections we recalculate the main coefficients of the above results and we summarize them in the Tables 1 and 2.
                                
                                We will follow the same notation as in <cit.>. Unfortunately it was not possible to use an analogous notation in the rest of the present paper. Anyway we will point out the different notations.
                                
                                
                                
                                
                                  §.§.§ Gevrey functions and dependency on γ
                                Assumption. Let α∈ [1/3,1), and let T, ℓ, γ be defined as in Assumption A5, <cit.>. 
                                
                                
                                (In the present paper, this corresponds to conditions (<ref>) and (<ref>)).
                                
                                
                                
                                Gevrey functions are used as smooth localizers in the constructions and 
                                their main properties are outlined in Section 4 of <cit.>.
                                In particular in our calculations we consider the following Gevrey function (see <cit.>, Ex 1.4.9 for definition):
                                
                                    χ_1(t) = χ(1+t)χ(1-t),   χ(s) = exp(-s^α/α-1)
                                    ,  χ(s)=0 .
                                
                                One can slightly modify the definition such that χ_1 = 1 in a ball B_1 ⊂  (with radius 1), χ_1 = 0 outside the ball B_2 (with radius 2), and 0≤χ_1 ≤ 1. Observe that χ_1 ∈ G^1/α_0() since
                                    |D^κχ_1(v)| ≤ c_0Xc^|κ|_1X |κ|^|κ|/α,
                                       c_0X=O(1),  c_1X=O(1/1-α).
                                
                                
                                
                                Here the symbol O (big-O) means
                                “comparable up to a an absolute multiplicative constant to”
                                (i.e. A = O(B) implies c_abs≤ A/B ≤ C_abs, for some positive numbers c_abs, C_abs). 
                                
                                
                                
                                Furthermore,
                                
                                define  χ_δ(v) := χ_1(v/δ), v ∈^M.
                                
                                
                                
                                Hence, ℱ_v →ζχ_δ(v) = δ^M ℱ_v →δζχ_1(v) for
                                
                                 ζ∈, and calling c_2X = 1/(eMc_1X)^α we get
                                
                                    |ℱ_v →ζχ_δ(v)|    ≤   δ^M c_0Xexp(δ H_B_2(ζ)-c_2Xδ^α |ζ|^α)·(χ_1),dv).
                                
                                
                                Product estimate: for v ∈ B_2(^M), calling c_0X,l,c_1X,l (resp. c_0X,m,c_1X,m) the coefficients in (<ref>) for χ_l (resp. χ_m),
                                
                                    |D^κχ_l(v)χ_m(v)|    ≤    c_0X,lc_0X,mmax{c_1X,l,c_1X,m} (max{c_1X,l,c_1X,m})^|κ||κ|^|κ|/α.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                We start by linking γ  and the coefficient (1-α), since both quantities tend to zero.
                                
                                Assumption. We assume θ = 1/2. Next, from now on the symbol
                                ∼ means “comparable up a multiplicative coefficient independent of γ or (1-α)  to”. 
                                
                                (i.e. A ∼ B implies c ≤ A/B ≤ C, with c,C independent 
                                 of  γ or (1-α)). The multiplicative coefficient is in general a uniform geometric constant, in the sense specified at the beginning of the paper. 
                                
                                
                                
                                We will call c_205 the resulting multiplicative coefficient for c_206.
                                
                                θ is the exponent appearing in the global stability of the unique continuation (see  Theorem <ref> of this paper and the following section <ref>), while 1/α is the order of the used Gevrey functions.
                                According to <cit.> (end of page 6469), by construction these two values are related in the following way:
                                
                                
                                    α^N =θα^1/r^(n+1) =  1/2  ⇒ (1- α) ∼ r^n+1α→ 1,
                                
                                where N=c_170∼γ^-58(n+1) 
                                
                                is reported in the following Table 2 and  r  ∼γ^58 is in Table 1. 
                                Consequently, for χ_1 and c_1X defined above, we get
                                
                                    c_1X∼1/1-α∼1/γ^58(n+1),    
                                    |χ_1'|_C^0(Ω_0)∼ c_1X,      |χ_1”|_C^0(Ω_0)∼ c_1X^2.
                                
                                
                                  §.§.§ Tataru inequality and Table 1
                                
                                
                                We consider the wave operator in ^n+1,
                                
                                    P(y,D) = -D_0^2+ ∑_j,k=1^n g^jk(x)  D_j D_k +  ∑_j=1^n h^j(x) D_j + q(x),
                                
                                where  y=(t,x)∈×^n are the time-space variables, D_0 = -i ∂_t, D_j = -i ∂_x_j.
                                The  coefficients g^jk∈ C^1(^n) are real and independent of time, and  [g^jk] is a symmetric positive-definite matrix.
                                The  coefficients h^j, q∈ C^0(^n)
                                are complex valued and independent of time. 
                                
                                
                                Call ξ=(ξ_0,ξ̃) the Fourier dual variable of y=(t,x).
                                
                                
                                  In the next theorem we use the exponential pseudodifferential operatore^-ϵ |D_0|^2/2τ v= ℱ^-1_ξ_0 → te^-ϵξ_0^2/2τℱ_t'→ξ_0v, with ℱ and ℱ^-1 representing respectively the Fourier transform and its inverse. 
                                
                                
                                Let us also define
                                
                                    f(y)= ∑_|υ|≤ 2 (∂^υϕ)(y_0)  (y-y_0)^υ / υ! - σ |y-y_0|^2.
                                
                                In following theorem (called Theorem 2.1 in <cit.>) we recall the Carleman-type estimate by Tataru, named `Tararu inequality'.
                                
                                (<cit.>, Theorem 2.1; or <cit.>, Theorem 2.3.) Let Ω be an open subset of ×^n.
                                Let P(y,D) be the wave operator (<ref>), with g^jk(x) ∈ C^1(Ω), h^j, q ∈ C^0(Ω).
                                Let y_0 ∈Ω and ψ∈ C^2,ρ(Ω) be real valued, for some fixed ρ∈(0,1), such that ψ'(y_0)≠ 0 and S={y;ψ(y)=0} being an  oriented hypersurface non-characteristic
                                in y_0. 
                                
                                Consequently there is λ>1 such that ϕ(y)=exp(λψ) is a conormally strongly pseudoconvex  function with respect to P at y_0.
                                
                                
                                Then there is a real valued quadratic polynomial f defined in (<ref>) with proper σ > 0,
                                and a ball B_R_2(y_0) such that f(y) < ϕ(y) when y ∈ B_R_2-{y_0} and f(y_0)=ϕ(y_0); and f being a conormally strongly pseudoconvex function with respect to P in B_R_2.
                                This implies that there exist ϵ_0, τ_0, c_1,T, c_2,T, R, such that, for each small enough  ϵ<ϵ_0 and large enough τ > τ_0, we have
                                
                                    e^-ϵ |D_0|^2/2τ e^τ f u_1,τ≤ c_1,T τ^-1/2e^-ϵ |D_0|^2/2τ e^τ f P(y,D)u_0 + c_2,T e^-τ R_2^2/4ϵe^τ f u_1,τ.
                                
                                Here u ∈ H^1_loc(Ω), with P(y,D)u ∈ L^2(Ω) and supp(u) ⊂ B_R(y_0).
                                Assumption:  We now consider the  `hyperbolic function' 
                                
                                    ψ(t,x;T,z) = (T - d_g(x,z))^2 - t^2
                                
                                introduced in Definition 3.1 of <cit.>, and its level set ψ(y) - γ^2 = 0.
                                
                                
                                
                                
                                
                                
                                Starting from  a 
                                
                                
                                  generalψ, in section 3 of <cit.>, page 180, we
                                have already calculated all the geometric constants associated either with the related pseudo-convexity estimates of ψ or with the Tataru inequality. 
                                They are summarized in Table 1, page 191 of <cit.>, and are copied in Table A.3. of <cit.> 
                                
                                (with few modifications explained in the related Appendix A).
                                Then in section A.1.1. of <cit.>, page 6487, we have recalculated these quantities for the particular case of  the `hyperbolic function' ψ in (<ref>). 
                                
                                
                                
                                Our aim here is to start from Table A.3. and section A.1.1. of <cit.>  in order to 
                                 find the γ-dependency of those coefficients.
                                
                                The following new Table 1 must be read from the top to the bottom, since it starts with the basic inequalities and continues with more complicated expressions.
                                
                                As said, we assume ψ as in (<ref>), and calculate all coefficients in the Tataru inequality.
                                The first two values C_l =min|ψ'(y)| and p_1=min p(y, ψ') are defined at page 6484 of <cit.>,  Section A.1., Assumption b).
                                Their limit value is calculated in <cit.>, formula (A.7): i.e. 
                                C_l = 2 γ_I b_0^-1/2,  p_1=4γ_I^2.
                                Since γ_I=γ/√(2) (see Lemma A.3.a, page 6489), and b_0 is defined as a constant (see formula (3.1), page 6452), then 
                                the γ-dependency of the two coefficients is respectively γ and γ^2, as shown in the table. 
                                The third value of Table 1 is dist{∂Ω_0,  Ω_a} (alias dist(Λ, ∂Ω_0)) and behaves like γ^2, thanks to the estimates (A.12) and (A.11) in <cit.>.
                                On the other hand, the following coefficients until C_3 in Table 1  are independent from γ, because of formula (A.6) and (A.8) in <cit.>.
                                
                                The next  values in Table 1 are obtained by substituting the upper values:
                                M_P, M_1,..., R_1, defined in section 3.1 of <cit.>;
                                c_T ∼λ^3 (replacing 4n|λψ|_max,Ω_0), see (A.2) and Remark A.1 in <cit.>;
                                τ_0, c_1,T, c_2,T, c_133 defined in section 3.2 of <cit.>;
                                r, δ, R, defined in section 3.3 of <cit.>, here we have renamed r_0 by r.
                                
                                These 3 coefficients are used to prove Proposition 2.5 of <cit.>, which is related to the result of local stability for the unique continuation.
                                c_111 of <cit.> is not used here.
                                
                                Note that σ, r, δ, R, τ_0, R_1, R_2 have nothing to do with quantities with the same name used in the rest of this paper (outside from the Appendix). 
                                
                                
                                
                                
                                 
                                
                                
                                  §.§.§ Global stability coefficients and Table 2
                                
                                This section can be seen as an overview of the proof of Theorem <ref>, with the final estimate for c_206.
                                
                                We introduce the main steps and we always follow the notation of <cit.> to better follow the calculations.
                                
                                
                                Assumption:
                                Define a net of center points (t_k,z_k) for the translated hyperbolic functions:
                                
                                    ψ(y;T_k,z_k,t_k) = (T_k-d_g(x,z_k))^2-(t-t_k)^2.
                                
                                Let Υ=W(z,T,ℓ) be the initial cylinder  (called Γ in (<ref>)) 
                                 and let Σ(z,ℓ,T) be the related domain of influence (in the paper called Σ(z,0,T) according to (<ref>)).
                                
                                
                                We choose the domains for the covering 
                                
                                    Ω_0,k⊂{y; y∈ [-T_k +t_k, T_k+ t_k ]×^n; ψ(y;T_k,z_k,t_k) ≥γ_k^2/2,  T_k≥ d_g(x,z_k)}
                                 and 
                                
                                    Λ_k⊂{y; y∈ [-T_k +t_k, T_k+ t_k ]×^n; ψ(y;T_k,z_k,t_k) ≥γ_k^2,  T_k≥ d_g(x,z_k)}.
                                 Let γ_k ≥γ, for all k.
                                
                                
                                The construction is similar to the one in Figure 1, page 6470 of <cit.>. 
                                
                                The parameters (t_k,z_k,T_k,γ_k) should be chosen such that the x-projection of Ω_0,k is  contained  in the domain 0 < d_g(z_k,x) ≤7/8i_0, that is within the injectivity radius i_0, in order to guarantee the C^2,ρ-regularity of ψ(y;T_k,z_k,t_k).
                                Moreover 
                                
                                the union Λ= ⋃_k=1^K Λ_k should cover a subset of the domain of influence Σ(z,ℓ,T).
                                
                                For example, let Λ = S(z,ℓ,T,γ), (alias (z,γ,T)  in (<ref>)).
                                
                                
                                
                                The above construction together with the assumption on the Gevrey-regularity of the localizers 
                                let us apply Theorem 1.2 in <cit.>. The details of Assumptions A2-A3 can be checked in the paper, while P is defined in (<ref>).
                                
                                (<cit.>, Theorem 1.2)
                                Under the conditions of Assumptions A2-A3,
                                
                                define the open set
                                Ω_1 = ⋃_k=1^K Ω_0,k\Υ containing Λ.
                                
                                
                                
                                Then for every 0<θ < 1 we have
                                
                                    u_L^2(Λ)≤ c_161u_H^1(Ω_1)/(ln(1+u_H^1(Ω_1)/Pu_L^2(Ω_1)))^θ.
                                
                                Moreover, for any m ∈ (0,1] we get
                                
                                    u_H^1-m(Λ)≤ c_161^m u_H^1(Ω_1)/(ln(1 + u_H^1(Ω_1)/Pu_L^2(Ω_1)))^m θ .
                                
                                The constant c_161 is calculated in the proof.
                                
                                
                                
                                Up to a uniform multiplicative constant (and according to Remark 3.8. of <cit.>), we can identify the constant c_161 with our final constant c_206,  even if the first one is defined for a bounded domain of the Euclidean space and the second one is defined for a compact manifold (M,g). Indeed by assumption, in each chart of M holds the inequality 
                                
                                    a_0 I≤ [g_jk(x)]_j,k=1^n≤ b_0  I,  and  g_jk_C^4(M)≤ b_3,   a_0<1<b_0,
                                
                                which let one approximate all spatial subdomains to an Euclidean ball.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                Theorem 1.2 is a generalization of Theorem 1.1 in <cit.> for a more complex domain, but with a similar final estimate where the inverse-log term has a different multiplicative constant.
                                For each  Ω_0,k Theorem 1.1 in <cit.> holds with constant c_160 in place of c_161.
                                The number 
                                
                                K of the used setsΩ_0,k is by construction proportional to the number of charts covering the domain. This number depends on 
                                 
                                 
                                 the bounds for the diameter, the injectivity radius and the harmonic radius  of M, called respectively D, i_0 and  r^(har) in the notation of the paper.
                                 
                                Hence we can also write c_206∼ c_161∼ c_160.
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                The technique used to prove the above Theorem consists in iterating the local stability result, but considering the low temporal frequencies separately from the high temporal frequencies.
                                The pseudodifferential operator A(D_0/ω) defined below is used to localize the low temporal frequencies of the solution u, where the estimate is more complicated.
                                
                                Assumption:
                                We consider a pseudo-differential operator A(D_0)  with symbol a(ξ_0) ∈ G^1/α_0(), 0≤ a≤ 1,
                                supported in |ξ_0|≤ 2 and equal to one in |ξ_0|≤ 1.
                                Hence we can write A(β |D_0|/ω)v = ℱ^-1_ξ_0 → ta(β |ξ_0|/ω)ℱ_t'→ξ_0v.
                                
                                We fix a as in (<ref>).
                                
                                
                                
                                
                                
                                Another complication comes from the fact that the local stability result holds just in small balls B_r(y_j), centered in y_j with radius r.
                                 
                                It is important for our estimates that the balls B_r(y_j), with j=1,…, N,  cover the set   Λ. 
                                We will choose the center points y_j in the set E, so that the union of the balls is contained in the domain of influence of the cylinder Υ, i.e. ⋃_k=1^N B_r(y_j) ⊆⋃_k=1^K Ω_0,k⊂Σ(z,ℓ,T).
                                
                                
                                 Furthermore there are particular  conditions on the support of u to be fulfilled, 
                                  also affecting the set E and the iteration.
                                
                                
                                Hence in the final domain ⋃_k=1^K Ω_0,k the local stability result must be applied several times to a sequence {u_j}_j=2^ N of proper cut-offs of the solution u.
                                Let u_j be defined as:
                                
                                    u_j = ∏_k=1^j-1(1-b_k)u,    b_k:=b(2(y-y_k)/r).
                                
                                
                                Then we can introduce the following Theorem 2.7 in <cit.>, formulating a local stability estimate (of the unique continuation for the wave operator) of inverse exponential type for the low temporal frequencies of u_j.
                                
                                
                                The exact construction of the radii r and R is in Proposition 2.5 of <cit.>, as intersection of several geometric and analytic constraints.
                                The γ-dependency of r and R is shown in Table 1.
                                
                                In particular we get r ∼γ^58.
                                The number of balls used in the iteration is N=c_170∼γ^-58(n+1), as shown in Table 2.
                                The constant c_170 is defined in formula (2.5) of <cit.>.
                                
                                
                                Notice that at each step we reduce the support of the temporal localizer A(D_0), by defining the term μ_j = c_156μ_j-1^α.
                                 We will show that 
                                c_155,N∼γ^- ζ_4 c_155,1∼γ^- ζ_5, 
                                and that c_161∼ N γ^- ζ_6  c_156^-α/(1-α),
                                for proper positive numbers ζ_4, ζ_5, ζ_6.
                                
                                
                                The details of Assumptions A1-A2-A4 and of the set E can be checked in the paper.
                                
                                (<cit.>, Theorem 2.7)
                                
                                
                                Under the Assumptions A1-A2-A4,
                                
                                
                                let y_k ∈ E
                                 
                                
                                and let
                                b∈ G^1/α_0(^n+1) be a Gevrey functions of class 1/α with compact support, such that 0<α<1.
                                
                                
                                Then, there exist constants R,r with R ≥ 2r >0, and c_159>1
                                
                                such that
                                 for μ > c_159 there are coefficients c_151, c_152, c_154,c_155,c_156, β, N for which,
                                
                                if
                                
                                    u_H^1(Ω_1)= 1,  Pu_L^2(Ω_1)
                                    
                                    < 1
                                    
                                    ,  A(D_0/βμ)l(y)Pu_L^2≤exp(-μ^α),
                                
                                then calling  μ_1=μ and μ_j = c_156μ_j-1^α for 2 ≤ j≤ N, we have μ_j ≥ 1 and
                                
                                    u_j_H^1(B_2R(y_j))≤ c_152,  Pu_j_L^2(B_2R(y_j))≤ c_153,
                                
                                    A(D_0/μ_j)b((y-y_j)/R)Pu_j_0 ≤ c_154,jexp(-μ_j^α),
                                
                                and consequently
                                
                                
                                    A(D_0/ω)b((y-y_j)/r)u_j_H^1≤ c_155,jexp(-c_132μ_j^α^2),  ∀ ω≤μ_j^α/(3c_131).
                                
                                The radii r and R are defined in Table A.3, while the coefficients c_k are calculated in the proof of the Theorem.
                                
                                
                                In the following Table 2 we show the γ dependency of the coefficients used in the proof of Theorem 1.2 and Theorem 2.7 in <cit.>.
                                
                                
                                
                                The coefficients c_h of the local stability are defined in <cit.> and recalled also in the proof of Lemma 2.6. of <cit.>, page 6459.
                                As said, the index h is unique and here we briefly remind the definition of c_h and the relationship with other coefficients and with Table 1.
                                
                                
                                In (<ref>) we obtained the γ dependency for  c_1X, i.e.  c_1X∼γ^-58(n+1).
                                It follows that c_2X∼ 1/(c_1X)^α, where c_2X is the coefficient in (<ref>) (it was called c_102 in <cit.>).
                                
                                Therefore for simplicity 
                                
                                
                                 we give below
                                
                                the values in Table 2 in terms of their c_1X or  γ dependency. 
                                
                                In order to calculate the rest we need to refine some estimates.
                                
                                First of all we recall and improve the coefficients in Lemma 2.1, <cit.>, for the L^2 and H^m norms:
                                
                                      c_107=  c_3 (8/β_1Γ(1/α)1/α (c_117)^1/α)^1/21/(α c_106)^1/α,
                                      
                                c_108=  c_107(1+|D_x^m f|_C^0) + c_107(1+m)^(m+1)/α/(α c_106)^m/α
                                      A(β_1 D_0/μ)f (1-A(D_0/μ))v_1 ≤ c_108 e^-c_106μ^αv_(f)_m . 
                                Next, following Remark 2.8 (4) in <cit.>, we split each smooth Gevrey localizer in time and space:
                                
                                    b(y-y_0/R)=b(t-t_0/R)b(x-x_0/R),
                                
                                with b(t) = χ_1(t) ∈ G^1/α_0() (as in (<ref>)) and b(x) ∈ C^2_0(^n).
                                Consequently the functions f_1(y),f_2(y),f_3(y) (see formula (2.21) in <cit.>) can generally be written as:
                                f_*(y)=f_*(t)f_*(x), with f_*(t) = D_0^2 b_j-1(t)+D_0 b_j-1(t)+b_j-1(t) and
                                f_*(x) = D_rD_s b_j-1(x)+D_r b_j-1(x)+b_j-1(x), for b_j-1(t) := b(2(t-t_j-1)/r).
                                Let v = b((y-y_j-1)/r)u_j-1, then
                                
                                    A(3D_0/ν)f_*(t)(1-A(D_0/ν))f_*(x)v_1
                                    
                                    ≤A(3D_0/ν)(D_0 f_*(t))(1-A(D_0/ν))f_*(x)v_0
                                    
                                    + A(3D_0/ν)f_*(t)(1-A(D_0/ν))(D_0+D_x+1)(f_*(x)v)_0
                                    
                                    ≤ c_108 c_152exp(-c_106ν^α)
                                
                                with c_108 calculated as in (<ref>) with β_1=3, m=3, c_3=(r/2) c_0X. Moreover, we can 
                                recalculate the terms at page 6466 of <cit.>:
                                
                                    c_162,j   =    2 c_162,j-1+ c_153c_164+c_155,j-1 |-P_2 b_j-1 + h^s(x)D_x_sb_j-1|_C^0
                                          +c_107 c_152(1+ n^2|g^kr|_C^0  + |h^s|_C^0)
                                    
                                     + c_155,j-1 |2  D_0 b_j-1|_C^1
                                    
                                     + c_152 c_108
                                          + c_155,j-1  |D_0(2  D_0 b_j-1)|_C^0
                                    
                                    + c_152c_107
                                          +  c_155,j-1 |2n g^kr  D_k b_j-1|_C^1
                                    
                                    +c_152 c_108 n^2|g^kr|_C^1
                                          + c_155,j-1 |D_r(2  g^kr D_k b_j-1)|_C^0
                                    
                                    +c_107 c_152n^2|g^kr|_C^1
                                
                                    c_162,j   ∼    2 c_162,j-1 + ( N^2 c_1X^2/r^2)c_1X^3/2/r^1/2 + c_155,j-1(1+ |g^kr|_C^1+|h^s|_C^0)(|b'|_0/r+|b”|_0/r^2+|b'|^2_0/r^2)+
                                          + ( N c_1X/r) c_108 (1+ |g^kr|_C^1+|h^s|_C^0)
                                     ∼  c_162,j-1 + c_155,j-1c_1X^2/r^2
                                    
                                    c_154,j   =   c_162,j + c_153c̃_107∼ c_162,j + N^2 c_1X^3/r^2R^n  ∼ c_162,j∼  c_155,j-1c_1X^2/r^2
                                    
                                    c_116   ∼   γ^4 c_154,j^2 (N c_1X/γ^48)^4.
                                
                                By applying Lemma 2.6 in <cit.> with c_U=c_152, c_P=c_153, c_A=c_154,j, one obtains:
                                
                                    c_155,j=c_150(c_152,c_153,c_154,j) ∼ c_1X^3√(c_116)/γ^48∼N^2 c_1X^5/γ^46+58· 2c_155,j-1,
                                    
                                    c_156 =min( 1/18β c_131,c_132^1/α,c_165^1/α, c_106^1/α/3c_131)
                                    
                                    = c_106^1/α/3c_131∼γ^56α + 58(n+1)(α+1)+28.
                                
                                Now we can obtain the γ dependency of c_160 in Theorem 1.1 of <cit.>.
                                
                                Recalling (<ref>) (i.e. α^N = θ = 1/2 and (1-α) ∼γ^58(n+1)),
                                
                                we get c_159 = c_156^-1/α^N-1(1-α) > 1 and therefore:
                                
                                    c_159∼(1/γ^56α + 58(n+1)(α+1)+28)^1/2γ^58(n+1)
                                    
                                    =exp(-[56α + 58(n+1)(α+1)+28]/2γ^58(n+1)ln (γ)),
                                
                                    c_158   =    N c_155,N+ 3N c_131 c_152(1+ |b'|_C^0/r)c_156^-α/(1-α)∼ N c_131 c_152c_159^1/2,
                                    
                                    c_160   =   (ln (1+ e^c_159))^1/2+2^1/2 c_158∼ c_158.
                                
                                Hence, c_160 of Theorem 1.1 in <cit.> (and analogously c_161 in Th. 1.2.) fulfills the estimate
                                
                                    c_160≤exp(1/γ^c_200), 
                                     c_200 = 58(n+1) + 2.
                                
                                
                                We know that c_206∼ c_160.
                                
                                
                                 We denote byc_205 the uniform multiplicative constant that depends on 
                                 the uniform geometric parameters  T,i_0,D,r_0,R,n, named according to the notation of the rest of the paper. 
                                
                                The number c_205 also depends on θ, that for simplicity has been fixed here equal to 1/2. 
                                
                                
                                
                                 The above inequality gives an estimate for c_160∼ c_206, and thus we
                                 
                                
                                can conclude that c_206(,θ)=c_205 ( θ)exp(γ^-c_200).
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                
                                 Remark.
                                Please notice that there was a misprint in the paper <cit.>
                                both in the statements of Theorem 3.3. and in Corollary 3.9. 
                                However this misprint did not affect the calculations of the present paper (or the results in <cit.>).
                                
                                
                                Namely in Theorem 3.3., we have the following
                                erratum (in the denominator of the final inequality):
                                
                                    u_L^2(Λ)≤ 
                                    c_163u_H^1((Ω_1)/(ln(e + u_H^1((Ω_1)/f_L^2(Ω_1)))^θ.
                                
                                And this is the corresponding corrigendum (replace e with 1):
                                
                                    u_L^2(Λ)≤ 
                                    c_163u_H^1((Ω_1)/(ln(1 + u_H^1((Ω_1)/f_L^2(Ω_1)))^θ.
                                
                                In Corollary 3.9., we have erratum in (3.27), and corrigendum:
                                
                                    w_L^2(Ω_2\ W_1)≤ 
                                    c_166w_H^1((Ω_1\ W_0)/(ln(1 + w_H^1((Ω_1\ W_0)/C' w |_W_1_H^1(W_1)))^θ.
                                 Table 1 and Table 2. We next present the two tables that summerize the previous calculations. They show the γ dependency of the parameters. The name of the constants there is unique. 
                                The order of the parameters in Table 1 is always increasing in complexity, that is the parameters down may depend on the upper ones.
                                In general the same principle is followed also in Table 2, even if the relationships are more complex. For simplicity 
                                the values in Table 2 are expressed in terms of their c_1X or  γ dependency, where we recall that  c_1X∼γ^-58(n+1).
                                
                                
                                 Table 1
                                
                                Name        Order with respect to γ
                                C_l   ∼   γ     (<cit.>, formula (A.7))
                                p_1   ∼   γ^2     (<cit.>, formula (A.7))
                                dist{∂Ω_0,  Ω_a}   ∼   γ^2     (<cit.>, formula (A.12))
                                |ψ'|_C^k   ∼   1     (<cit.>, formula (A.8))
                                d_g(x,z)   ∈   [ℓ, T-γ]   (in Γ\ cylinder)
                                |∂_k d_g|   ∼   1     (<cit.>, formula (A.6))
                                C_3   ≥   1
                                M_P   ≤    1
                                M_1   ≥   1/(p_1)^2 = 1/γ^4
                                M_2   ≥   M_1= 1/γ^4
                                λ   ≥   max{M_1,e,1/C_l^2}= 1/γ^4
                                ϕ_0   ≥   e^-1
                                ϕ_M   ≤   e
                                R_1   ≤   min{1, γ^2, 1/λ}=γ^4
                                c_T   ∼   λ^3=1/γ^12    (<cit.>, formula (A.2)  and Remark A.1)
                                c_100     ≥   1
                                ϵ_0   ≤   1/(λ (1 + λ )+ c_T)=1/λ^3=γ^12
                                R_2   ≤   min{R_1,  C_l/ (1 + λ   + c_T/λ ), λ^2  C_l^2/c_T,
                                
                                (1/c_T^2  M_1(1+λ^2))^1/4, ϵ_0/√(2M_2), λ/c_T
                                (1+λ^2  +λ^2 (1 + λ) )},
                                
                                      =min{γ^4, γ^9,γ^6, γ^9, γ^14, γ^20}=γ^20
                                σ   ≥   c_T R_2=γ^8      (<cit.>, formula (A.2) and Remark A.1)
                                τ_0   ≥   M_1  (
                                (λ^2 +  c_T R_2)^2  + |h|^2_C^0(Ω_0)(1+(λ  + c_T R_2^2)^2)+ |q|^2_C^0(Ω_0)) = 1/γ^20
                                R   ≤   R_2 = γ^20
                                δ   ≤   c_T R_2^3=γ^48
                                r   ≤   λ^2 C_l^2 R_2^3/( λ+c_T R_2^2 ) = γ^58
                                c_1,T   ≥   √(( M_1/τ_0 + 1/λ))=γ^2
                                c_2,T   ≥   √(M_2)(1+|χ_1'|_C^0(Ω_0)/τ_0 R) + c_1,T/√(τ_0) c_133=1/γ^2+1/γ^8(|χ_1”|_C^0(Ω_0)+|χ_1'|_C^0(Ω_0)/γ^4 ) ∼c_1X^2/γ^8
                                c_133   ≥   |χ_1”|_C^0(Ω_0)/τ_0 R^2 + |χ_1'|_C^0(Ω_0)/R(1+λ  + c_T R_2^2+|h|_L^∞(Ω_0)/τ_0) =1/γ^20(|χ_1”|_C^0(Ω_0)+|χ_1'|_C^0(Ω_0)/γ^4 )
                                
                                
                                 Table 2
                                Name       Value     Name     Value 
                                c_2X   =c_102=1/(e c_1X)^α   c_119   δ c_1X∼γ^48c_1X
                                c_118   1 + |ϕ'|_0(1+R_2) +5n|ϕ”|_0,ρR_2^ρ+1 + |ϕ”|_0(1+R_2^2) +  σ(2+R_2^2)∼1/γ^8   c_114   c_1,T^2 |g|^2_C^1|χ_1|^2_C^2(1+|φ'|^4_C^0/δ^4+|φ”|^2_C^0/δ^2)
                                 ∼c_1X^4/γ^12+48· 4
                                c_115   c_2,T^2(|φ'|^2_C^0 +1)(3^3 e^-3/δ^3)(1+|χ_1'|^2_C^0/δ^2)
                                ∼c_1X^6/γ^8· 3+48· 5   c_121   c_1X/δ
                                c_122   c_1X^2/γ^44   c_123   ∼γ^56 ·α/c_1X^α
                                c_128   1/3^α2c_123∼ c_123   c_110   c_122(8Γ(1/α)/3[α c_123^1/α(α c_128)^1/α])^1/2∼ c_1X^3/γ^44+56
                                c_109   min(√(ϵ δ/36), c_128/2, 1)
                                 ∼γ^56 ·α/c_1X^α   c_130   3 c_109/4 δ(1/16 )^5  ∼γ^56 ·α - 48/c_1X^α
                                c_131   (16^6√(2), 16^6 3^α-1√(2ϵ_0 δ)/c_123, 16^6 √(ϵ_0 δ)/3√(2))  ∼c_1X^α/γ^56·α - 30   c_135   r^α c_2X1/4 · 3^α∼γ^58·α/c_1X^α
                                c_137   min(1/2(c_102δ^α(c_130)^α/(√(2))^α +δc_130/2 √(2)), 1/2 c_102δ^α(1/2√(2)c_130)^α)
                                 ∼γ^48·α/c_1X^αc_130^α   c_132   min(c_135,c_137)   ∼γ^56 ·α·α/c_1X^α·α1/c_1X^α
                                c_170   N ∼1/γ^58(n+1)   c_117   (r/2)^α1/(ec_1X)^α∼r^α/c_1X^α
                                c̃_117   c_2X R^α = (e c_1X)^-α  R^α   β   2 + (4/c̃_117)^1/α∼c_1X/R   (2.11)
                                c̃_106   1/β^α∼R^α/c_1X^α   c̃_107   R^n+1 c_0X(8/βΓ(1/α)1/α (c̃_117)^1/α)^1/21/(αc̃_106)^1/α∼ R^nc_1X
                                c_154,1   1 + c̃_107∼ R^nc_1X   c_155,1   max(c_134, c_136) = max(c_1X^2.5γ^58(n-3/2) , c_1X^6/γ^180)=c_1X^6/γ^180
                                c_153   1+2N(1+ n^2|g^kr|_C^0 + |h^s|_C^0)(|b'|_C^0/r + |b”|_C^0/r^2+ (N-1)|b'|^2_C^0/r^2) ∼N^2 c_1X^2/r^2   c_152   2(1+ N |b'|_C^0/r) ∼Nc_1X/r
                                c_162,1   1   c_156   ∼c_106^1/α/3c_131∼γ^56α + 58(n+1)(α+1)+28
                                c_165   c_117β^α/(3^α 4) ∼r^α/R^α∼γ^38α   c_164   r/2 c_0X(8/3Γ(1/α)ec_1X/α^1/α (r/2))^1/2e c_1X (3^α 4)^1/α/(α^1/α (r/2))∼c_1X^3/2/r^1/2
                                c_107   c_164∼c_1X^3/2/r^1/2   c_108   (c_107 + c_1074^4/α/(α c_106)^3/α)
                                (1+|b'|_0/r+|b”|_0/r^2+|b”'|_0/r^3) (1 + |b'|_0/r)
                                ∼c_1X^17/2/r^15/2
                                
                                1
                                Al Alessandrini G.  Stable determination of
                                conductivity by boundary measurements, Appl. Anal.,  27 (1988), 153–172.
                                
                                
                                AlS Alessandrini G., Sylvester J. 
                                Stability for a multidimensional inverse spectral theorem.
                                Comm. Part. Diff. Eq.  15 (1990),  711–736.
                                
                                And Anderson M.  Convergence and rigidity of manifolds
                                under
                                Ricci curvature bounds, Invent. Math.,  102 (1990), 429-445.
                                
                                AKKLT
                                Anderson M., Katsuda A., Kurylev Y., Lassas M., Taylor M.   Boundary regularity for the Ricci equation, Geometric Convergence, and Gel'fand's Inverse Boundary Problem,Invent. Math. 158 (2004), 261-321.
                                
                                Be
                                Belishev, M.  An
                                approach to multidimensional inverse problems for the wave
                                equation. (Russian) Dokl. Akad. Nauk SSSR,  297 (1987), 524–527
                                
                                BeKu2 
                                Belishev, M., Kurylev, Y. 
                                 A nonstationary inverse problem for the
                                multidimensional wave equation "in the large". 
                                (Russian) Zap. Nauchn. Sem. 
                                LOMI,  165 (1987), 21–30.
                                
                                BeKu Belishev M., Kurylev Y.  To the reconstruction
                                of a Riemannian manifold via its spectral data (BC-method), Comm. Part. Diff. Eq.,
                                 17 (1992), 767-804.
                                
                                BeBeG Berard P., Besson G., Gallot S.  Embedding
                                Riemannian manifolds
                                by their heat kernel, Geom. Funct. Anal.,  4 (1994), 373-398.
                                
                                BL
                                Bergh J., Löfström, J.  Interpolation spaces. An introduction. Springer-Verlag, 1976,  pp. x+207,
                                
                                Bl1
                                  Blagoveščenskii, A. A one-dimensional inverse boundary value
                                problem for a second
                                order hyperbolic equation. (Russian) Zap. Nauchn. Sem.
                                LOMI, 15 (1969),
                                85–90.
                                
                                B Bosi R., Kurylev Y.,  Lassas M.  Stability of the unique continuation for the wave operator via
                                 Tataru inequality: the local case, Journal d'Analyse Mathematique, Vol. 134 (2018), 157 – 199.
                                
                                BKL Bosi R., Kurylev Y., Lassas M.
                                 Stability of the unique continuation for the wave operator via Tataru inequality and applications, J. Differential Equations, 260, 8, (2016), 6451-6492.
                                
                                BuBuI Burago D., Burago Y. and Ivanov S.
                                 A Course in Metric Geometry. AMS, Providence (2001).
                                
                                Chavel  Chavel I. Riemannian geometry. A Modern Introduction, 
                                
                                2nd ed,
                                2006.
                                
                                Ch
                                Cheeger J., Finiteness theorems for Riemannian manifolds.  Am.
                                J. Math. 92 (1970), 61–75.
                                
                                Davies  E. Davies,
                                 Spectral Properties of Compact Manifolds and Changes of Metric.
                                 American Journal of Mathematics 112  (1990),  15-39.
                                
                                dH1
                                de Hoop M.,  Holman, S., Iversen, E., Lassas, M., Ursin B.    Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times. J.  Math. Pures et Appl.  103 (2015), 830-848
                                
                                
                                dH2
                                de Hoop M.,  Holman, S., Iversen, E., Lassas, M., Ursin B.  Recovery of a conformally Euclidean metric from local boundary diffraction travel times. SIAM Journal on Mathematical Analysis  46 (2014), 3705-3726
                                
                                DKSaU Dos Santos Ferreira D.,  Kenig C., Salo M., and Uhlmann G. Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), 119–171.
                                
                                FIKLN Fefferman C., Ivanov S., Kurylev Y., Lassas M., Naranayan H.
                                 Reconstruction and interpolation of manifolds I: The geometric Whitney problem.
                                Preprint arXiv:1508.00674
                                
                                Ge Gelfand I.  Some aspects of functional
                                 analysis and algebra. 1957 Proceed.
                                the Intern. Congr. Mathem., Amsterdam, 1954,  1, 253–276.
                                
                                GW
                                Greene, R., Wu, H.  Lipschitz convergence of Riemannian manifolds. Pacific J. Math.  131 (1988),  119–141.
                                
                                Gr Gromov M. with appendices by Katz M., Pansu P. and Semmes S.,
                                Metric Structures for Riemannian and Non-Riemanian Spaces, based on
                                `Structures metriques pour les varietes riemanniennes',
                                (LaFontaine J. and Pansu P. eds),
                                Birkhauser (1999).
                                
                                GS Guillarmou C., Sa Barreto A. Inverse problems for Einstein manifolds, Inverse Probl. Imaging   3 (2009),  1–15.
                                
                                Ivanov
                                Ivanov, S.  Distance difference representations of Riemannian manifolds. arXiv:1806.05257
                                
                                Helin
                                Helin, T., Lassas T., Oksanen L., Saksala, T. 
                                 Correlation based passive imaging with a white noise source. 
                                J.  Math. Pures et Appl.
                                 116 2018,  132–160.
                                
                                H1
                                 Hörmander L.   The analysis of linear partial differential operators I.
                                 Springer-Verlag, 1985, viii+525 pp.
                                
                                KasKum Kasue A., Kumura H.  Spectral convergence of
                                Riemannian manifolds,
                                Tohoku Math. J.,  34446 (1994), 147-179.
                                
                                KasKum2 Kasue A., Kumura H.
                                 /it Spectral convergence of Riemannian manifolds. II. Tohoku Math. J.  48 (1996), 71-120.
                                
                                KaKuLa Katchalov A., Kurylev Y., Lassas M.
                                Inverse Boundary Spectral Problems, Chapman/CRC, Boca Raton (2001).
                                
                                Ka Kato T. Perturbation Theory for Linear Operators, Springer,
                                Berlin (1995).
                                
                                Katsuda Katsuda, A. 
                                
                                Gromov's convergence theorem and its application.
                                Nagoya Math. J.
                                 100 (1985), 11–48.
                                
                                
                                
                                KatsudaKuLa
                                Katsuda A., Kurylev Y., Lassas M.  Stability of boundary distance representation and reconstruction of Riemannian manifolds.Inverse Problems and Imaging 1 (2007), 135–157.
                                
                                Ku1 Kurylev Y.   Multidimensional Gel'fand inverse boundary problem and
                                boundary distance map. In: Inv. Probl. Related to Geom. (H. Soga, ed.), 1-15, Ibaraki Univ. 
                                Press, Mito, 1997.
                                
                                KLY Kurylev Y., Lassas M., Yamaguchi T.
                                  Uniqueness and Stability in Inverse Spectral Problems for Collapsing Manifolds.
                                Preprint arXiv:1209.5875.
                                
                                KrKL
                                Krupchyk K., Kurylev Y., Lassas M.  Inverse spectral problems on a closed manifold. J. Math. Pures Appl.  90 (2008), 42–59.
                                
                                LaU
                                   Lassas  M.,  Uhlmann G.  Determining Riemannian manifold
                                from boundary measurements.Ann. Sci.
                                École Norm. Sup. 34 (2001), 771–787.
                                
                                LaLe Laurent C., Leautard M.
                                 Quantitative unique continuation for operators with partially analytic coefficients. Application to approximate control for waves.
                                (2015), arXiv:1506.04254. To appear in JEMS.
                                
                                Mandache 
                                Mandache N.  Exponential instability in an inverse problem for the Schrodinger equation.
                                Inverse Problems,  17 (2001):1435.
                                
                                NSU
                                Nachman, A. Sylvester, J., Uhlmann, G. 
                                 An n-dimensional Borg-Levinson theorem. 
                                Comm. Math. Phys.  115 (1988), 595–605
                                
                                Nv1
                                Novikov, R. 
                                 A multidimensional 
                                inverse spectral problem 
                                for the equation 
                                -Δψ +(v(x)-Eu(x))ψ=0. 
                                (Russian) Funk. Anal. i Prilozhen.  22 (1988), 11–22.
                                
                                
                                ONeill
                                O'Neill, B.  Semi-Riemannian geometry. Pure and Applied Mathematics, 103. Academic Press,  1983. xiii+468 pp.
                                
                                Pe Petersen P. Riemannian Geometry, 1st Ed., Springer, New York (1998).
                                
                                Robbiano
                                Robbiano, L.  Fonction de cout et controle des solutions des equations hyperboliques. 
                                Asymptotic Anal.  10 (1995),  95–115. 
                                
                                RD  Rodino L. Linear Partial Differential Operators in Gevrey Spaces, World Scientific, (1993).
                                
                                StU Stefanov, P., Uhlmann, G.
                                 Stability estimates for the hyperbolic Dirichlet to Neumann
                                 map in anisotropic media. J. Funct. Anal.  154 (1998),
                                330–358.
                                
                                SU Sylvester J., Uhlmann G.
                                 A global uniqueness theorem for an inverse boundary value problem.
                                 Ann. of Math. (2)  125 (1987), 153–169.
                                
                                Ta Tataru D.  Unique continuation for solutions to PDE's:
                                between Hormander's theorem and Holmgren's theorem,
                                Comm. Part. Diff. Eq.,  20 (1995), 855-884.
                                
                                Ta1 Tataru D.
                                 Carleman estimates, unique continuation and applications, preprint.
