This paper provides new sufficient and necessary conditions for the frame property of generalized translation-invariant systems. The conditions are formulated in the Fourier domain and consists of estimates involving the upper and lower frame bound. Contrary to known conditions of a similar nature, the estimates take the phase of the generating functions in consideration and not only their modulus. The possibility of phase cancellations makes these estimates optimal for tight frames. The results on generalized translation-invariant systems will be proved in the setting of locally compact abelian groups, but even for euclidean space and the special case of (composite) wavelet systems the results are new.
@cite_7: Abstract : We study in this paper certain types of bases for shift-invariant subspaces. Our primary objective is to connect among three important families of basis sets: shift-invariant sets. Weyl-Heisenberg sets, and affine (wavelet) sets. The present paper is the first in a series of three, and is concerned with the basic theory of shift-invariant bases for the shift-invariant spaces. The two papers, (RS1) and (RS2), will focus on the applications of the theory developed here to Weyl-Heisenberg and affine sets
@cite_28: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >
@cite_36: In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd.
@cite_9: We present formulations of the condition of duality for Weyl-Heisenberg systems in the time domain, the frequency domain, the time-frequency domain, and, for rational time-frequency sampling factors, the Zak transform domain, both for the one-dimensional time-continuous case and the one-dimensional time-discrete case. Many of the results we obtain are presented in the more general framework of shift-invariant systems or filter banks, and we establish, for instance, relations with the polyphase matrix approach from filter bank theory. The formulation of the duality condition in various domains is notably useful for the design of perfect reconstructing shift-invariant Weyl-Heisenberg analysis and synthesis systems under restrictions of the constituent filter responses which may be stated in any of the domains just mentioned. In all considered domains we present formulas for frame operators and frame bounds, and we compute and characterize minimal dual systems.
@cite_21: Let ( ) be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group (G ). In this paper we study translation-invariant (TI) subspaces of (L^2(G) ) by elements of ( ). We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that ( ) be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of (G= R ^n ). We also extend the notion of the spectral function in ( R ^n ) to the LCA setting. It is shown that spectral functions, initially defined in terms of ( ), do not depend on ( ). Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of (G ) with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.
@cite_6: Let (a>0, b>0, ab<1; ) and let (g L^2( R ). ) In this paper we investigate the relation between the frame operator (S:f L^2( R ) n,m ,(f,g_ na,mb ) ,g_ na,mb ) and the matrix (H ) whose entries (H_ k,l ,; ,k',l' ) are given by ((g_ k' b,l' a ,g_ k b,l a ) ) for (k,l,k',l' Z . ) Here (f_ x,y (t)= exp (2 iyt) ,f(t-x), ) (t R ), for any (f L^2( R ). ) We show that (S ) is bounded as a mapping of (L^2( R ) ) into (L^2( R ) ) if and only if (H ) is bounded as a mapping of (l^2( Z ^2) ) into (l^2( Z ^2). ) Also we show that (AI S BI ) if and only if (AI 1 ab ,H BI, ) where (I ) denotes the identity operator of (L^2( R ) ) and (l^2( Z ^2), ) respectively, and (A 0, ) (B< . ) Next, when (g ) generates a frame, we have that ((g_ k b,l a )_ k,l ) has an upper frame bound, and the minimal dual function (^ ) can be computed as (ab , k,l ,(H^ -1 )_ k,l ,; ,o,o ,g_ k b,l a . ) The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a (g ) generating a frame are inherited by (^ . ) In particular, we show that (^ S ) when (g S ) generates a frame (( S ) Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula.
@cite_3: 
@cite_2: Using the range function approach to shift invariant spaces in L2(Rn) we give a simple characterization of frames and Riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2(Zn). This in turn gives a simplified approach to the analysis of frames and Riesz families done by Gramians and dual Gramians. We prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator. As an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function.
@cite_25: 
For shift-invariant systems, i.e., generalized shift-invariant systems with a single, fixed translation lattice @math , the Bessel and frame properties can be in terms of (bi-infinite) matrix-valued functions, so-called , as introduced by Ron and Shen @cite_7 , see also Janssen @cite_9 . As a consequence, the aforementioned necessary and sufficient conditions in , and are easily derived For example, the necessary condition in for shift-invariant systems follows from the norm estimate @math by noticing that the @math th column the dual Gramian matrix at @math is @math . from simple norm estimates of bi-infinite Hermitian matrices @math on @math , see [Section 1.6] MR1350650 . In particular, the estimates and are known for separable Gabor systems @cite_6 @cite_25 ; these estimates are the best known improvement of Daubechies' Gabor frame bound estimates @cite_28 . Furthermore, the dual Gramian characterization has, in a fiberization formulation @cite_2 , been extended to the setting of locally compact abelian groups @cite_21 @cite_36 @cite_3 . Hence, for shift-invariant systems, or more generally, trans -la -tion-in -vari -ant systems on such groups, the conditions , and follow from these characterizations and should not be considered new.