Applied Mathematics, 2011, 2, 1076-1090 doi:10.4236/am.2011.29149 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Numerical Solution of Mean-Square Approximation Problem of Real Nonnegative Function by the Modulus of Double Fourier Integral Petro Savenko, Myroslava Tkach Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine E-mail: spo@iapmm.lviv.ua, tmd@iapmm.lviv.ua Received May 27, 2011; revised June 21, 2011; accepted June 29, 2011 Abstract A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are pre-sented. Keywords: Mean-Square Approximation, Discrete Fourier Transform, Two-Dimensional Nonlinear Integral Equation, Nonuniqueness and Branching of Solutions, Two-Dimensional Nonlinear Spectral Problem 1. Introduction A variational problem about mean-square approximation of a real finite function by the modulus of double Fourier integral with use of smoothing functional  is studied. The nonuniqueness and branching of solutions is an es-sential feature of nonlinear approximation problem. The problem of finding a set of branching points is insuffi-ciently investigated nonlinear two-parameter spectral problem. The existence of connected components of the spectrum, which in the case of real parameters, similarly as in , are spectral lines, is essential difference of two-dimensional spectral problems compared with one- dimensional ones. The algorithms for finding the lines of possible bran- ching of solutions of the Hammerstein type nonlinear equation, which are based on implicit functions methods, are proposed and justified. The algorithms for numerical finding the optimal solutions of the approximation prob- lem are constructed and justified also. Numerical exam- ples are presented. Note that this class of problems are widely used at solving the inverse problems of radio physics, acoustics and so on [3,4]. 2. Problem Formulation, Basic Equations and Relations Consider the linear integral operator  12112 2,,exp ddGfss AUUxyicxs cysxy, (1) which is the double Fourier transform of function 2,UxyL G, dependent on the real two-dimen- sional parameter 12,ccc, 0ic1, 2i1. Operator acts from space into the solid angle U2LG2L, where 2 is some limited domain in which a real continuous nonnegative and nonzero function 12,Fss is given. In the spaces 2LG and 2L we introduce scalar products and generable by them norms 2222121 212124π,,,,LGGLGLGUUUxyUxy xyccUUU,dd (2) 1Parameters c1, c2 are physical parameters of the object being investi-gated. In particular, in the antennas synthesis problems these parame-ters characterize the electrical sizes of aperture of radiating system and a solid angle in which the necessary energetic characteristic of radiation is given . P. SAVENKO ET AL.1077 21211221212,,,Lddfffssfss ss, 212,Lfff. (3) Since the domain is limited, the integral in (1) exists in the usual sense  for an arbitrary function . Here the function 2G2ULG12,fss is continuous and quadratically integrable. Consider the problem about approximation of a real continuous and nonnegative function 12,Fss in the domain  by the modulus of the Fourier integral (1). We shall formulate it as a minimization problem of the smoothing functional   2222222()22min.  LGLUL GLGLUFAU UFf U (4) Here the first summand describes the mean-square de-viation of the modulus of the Fourier integral from the given function 12,Fss in the domain . The second summand imposes constraints on the norm of the Fourier integral prototype,  is a weight (regularizing) pa-rameter. Equating the Hato differential of the functional (4) to zero and taking into account (1), we write the equation concerning the function U that describes the fixed points in the space : U2LGargiAUUAAUAFe , (5) where  11221212 122,e dd2πicxs cysccAffss ss is conjugate operator with A. Further we introduce to shorten records the following notations: 12,Qss, , , . 12dddQss,PxydddPxyTaking into account that a set of zeros NA con-sists only of zero element, and acting on both parts of (5) by operator A, we obtain equivalent to (5) equation with respect to function 12,fss in the space G2Lif argfAAfAAFe. (6) Accordingly to the introduced above notations this equation in the expanded form takes the form arg (),, ()d,,e d,  ifQfQf KQQfQQKQQ FQQBcc (7) where  121112222,,expd d(2π)GKQQcc icxs scyssxyc (8) is a kernel dependent on the form of the domain . In the case of symmetric domain (8) can be simplified. In particular, if the axis is the axis of symmetry of the domain and its upper and lower limits are de-scribed, respectively, by the functions GGOXGyx at 1,1x , the kernel (8) is real and it has the form 122211112221,,sincosd .2πKQQcs sxccx ssxssc(9) Lemma 1. Between solutions of Equations (5) and (6) there exists bijection, that is if is the solution of (5) Uthen fAU is the solution of (6). On the contrary, if f is the solution of (6) then 11exp aUAf Ff rgi is the solution of (5). Proof. Let U be a solution of (5). Then arg 0iAUUAA AFe 11U. Acting on this equality by the operator A, we obtain arg11iAUAUAAAA Fe 0AU. The ope- rator acts from the space 2LG into the space 2L, and a set of its zeros consists only of zero element. Then from the last identity follows that 2AU fL is a solution of the equation arg11 0iffAAfAAFe , that is (6). On the contrary, let 2fL be a solution of (6). The operator A acts from the space into the space 2L2LG , and the Hilbertian space 2L coin-cides with the space . From here follows, that 2LA acts from the space into the space 2LG2L. Taking into account that F and f are continuous functions, the function exp argFif is quadrati-cally integrable in the domain . From here follows 112exp argfFifL. Thus, 112exp argAfFifUL  G and the right part of (6) is a result of action of operator A on the element U, that is 11exp argAUAAfF i ff  . Thus owing to the fact that AUf, we write this equality in the form 11exp arg0AUAAUAFiAU  . Since a set of zeros of operator A consists only of Copyright © 2011 SciRes. AM P. SAVENKO ET AL. 1078 zero element we have 11exp argUAAUAFiAU  . So, (10) 11exp argUAfF i f solves (5). Lemma is proved. Using the general expression (8) for the kernel ,,cKQQ,, dcDfAAfKQ QfQQ we shall consider a self-adjoint operator (11) and corresponding to it quadratic form for arbitrary func-tion :  2fQ L  2222,,,dexp,d d0.2π  GDffKQQf QQf QQcc fQiPQQPccd This equality to zero is achieved only as 0fQ. From here follows that the operator is nonnegative in  and, respectively, in D2LC. Based on this property, the operator retains the nonnegative functions cone DCK invariant, that is . DKKSince a set of values of operator A is a set of con-tinuous functions , belonging to the space 2L, and a set of continuous functions in the domain , is dense in the space 2 , we shall investigate the Lsolutions of (6) in the space . L2On the basis of decomplexification  we consider the complex space  as a direct sum  CC  of two real spaces of continuous functions in the domain . The elements of this space have the form: ,Tfuv, ReufC, ImvfC. Norms in these spaces we shall introduce as:  max,max ,max, . CCQQCCuuQvvfuvQ (12) The Equation (6) in decomplexified space  we reduce to equivalent to it system of equations   11111121122122,,,,,,uBuvBu BuvvBuvBvBuv (13) where 11 ,, dcBKQQuQ Q,   12 22,, dcuQBFQKQQuQ vQ 21 ,, dcBKQQvQ Q,   22 22,, dcvQBFQKQQuQ vQQ . (15) Note, are linear integral operators and 11B21B11 21BB. Denote a closed convex set of continuous functions as MS supposing that ,::,  uvu uvvMM MMMCMMCSSSSuS uqMSvSvqM, where  1111 , CCqI B max, ,dcQMFQ KQQQ  , (16) I is a unit operator in ()C. We show that operator 12 determined by ,BTBB(13) acts in the space Q; (14) . At first consider ,Buv1. The first component of this operator, defined by (14), is a linear integral operator with the kernel ,,cKQQ which is continuous on both arguments. Consequently, 11 :BCC is a continuous operator . Show that 12 :B. Let ,Tfuv be an arbitrary function belonging to . For 0ic 1, 2i the kernel is a con-,,cKQQtinuous function with respect to its arguments in the closed domain . Then according to the Cantor theorem  ,,cKQQ is a uniformly continuous function in . From here follows: at fixed for any points c11,QQ and 22,QQ such that whenever 112 2,,Q QQQ, then  112 2,,, ,ccKQQKQ Qa, here daFQQ . On this basis we have   121222,, ,,dduQ uQFQKQQKQ QuQ QuQ vQFQ Qa cc (17) since  22max 1QuQuQ vQ. Thus, 12 ,uBuv Copyright © 2011 SciRes. AM P. SAVENKO ET AL.1079 is a continuous function and 12 :()()B . Ana- logously we show that 22 :()()B. Consider corresponding to (13) linear homogeneous equation  ,, dcuQKQQuQ Q . (18) Above it is shown that the integral operator in the right part of (18) is self-adjoint and positively determined. Hence, its eigenvalues are real and nonnegative . From here follows that cannot be eigenvalue of (18). Then this equation has only zero solution uQ()0. Thus it is shown that necessary and sufficient condition for existence of inverse operator is satis- 1111IBfied . Since 2111 , then BB1121IB exists too. Easily to show  that 21111111111 11,0,sup 1,uLuIBIBuIBuquu. From here follows, that is a limited operator. 1111IBUsing existence and limitations of the operators and 11IB111121IB11v Iw1222,. (19) e write (13) in the form 1111111221,,,,uBuB BuvvBuvIB Buv Theorem 1. The operator determined 12,BTBBby (19) maps a closed convex sanach space et SM of the B() in itself and it is completely continuous. Proof. Before it was shown that :()B(). To prove the property of completee continuity of thoperator 12,BTBB it is necessary to prove its com- pactness y . We consider each of opera-and continuittors 1,Buv and 2,Buv in a system of Equations (19) aduct ofmited (continuous) and non- s the pro linear lilinear operators. Since 11IB, 11IB11 21 are limited operators, then for complete continuity of the operator 12,BTBB it is sufficient to show complete continuitys of operator12 ,Buv, 22 (,)Buv. We shall show it on the example 12 ,v.Bu Let 111,Tfuv and, T222fuv be arbitrary fun- ctions belonging toMS, and or 10v. It is 10unecessary to show, that 12 1CBf Bf12 20s a12 0ff. Let us  assume v2 = v1 + Δv Takinain2 21uu u,g into account these equalities we obt21uuu22 2 222221 111 2211221uvuuvv uvuv uv  At . 0Cu, 0Cv we have: 1222 220,112201220,110221122112222 1111 2211limlim max112210,221    uvuQvuuuvuvuuvuu vvuvuvuuu vvuvuv uv(20) since 2211220, 11022lim max11uQvuu vvuvuv  . Analogously we obtain 1222 220,11 220lim max0uQvvvuv uv . (21) Thus, from equalities (20) and (21) follows    121 112220,00,01222 2211 22lim max,,d0uvuQvFQ KQQuQu QQuQ vQuQ vQ lim ,,Buv Buv  c. Analogously ()()221 122220,0lim ,,0CCuvBuvBuv. Therefore, is a continuous operator fro12,BTBB m  into . Sho a setunw that of fctions satisfies 12wMSBS2Further for reduction of notations the dependence of functions u1, u2, Δu, Δv on the variable Q is omitted in (22) and (23) . co, that is nditions of the Arzela Theorem we show Copyright © 2011 SciRes. AM P. SAVENKO ET AL. 1080 that functions of a set wS are uniformly bounded and equapotentially continuoLet 12 ,wB uv, where us. ,Tfuv is an arbitrary function of a set MS. Then for 12,QQ ,QQ analogously w(17) with e have 12 dwQwQFQQa . Thus the functions of a set are equapoten-12wMSBStially continuous. Uniform boundedness of a set follows fr 12wMSBSom inequality     2222max,,d()max,,d,QQuQFQ KQQ QuQ vQuQwFQKQQQMuQ vQ    cc(22) where (,)Tfuv is an arbitrary function of a set SM. Thusr 1B is completely continuous in the the operato first equation of system (19). Complete continuity of operator 2B is proved analogously. Let ,)T(fuv be an arbitrary function of a set S anMd ( ,)T,) (TghuBv. Show that the function (,)Tgh belongs to a set MS. Using the inequalities AxAx and 111IB q 11 we have   11111() (,).CCCCCCgQB uvIBw  qMAnalogously we obtain:    21121(,)hQB uv.CCCCCCIBqM   From these inequalities follows that BMMSStely contin. So, the operator 12,BTBB is compleuous mapping the cloet sed convex s()MS into itself. The theorem is proved. As a corollary of Theorem 1 satisfaction of the Schauder principle conditions  in accordance to which the operator 12,BTBB has a fixed point ,Tfuvto a set , belonging MS, foll(13) and, respectively). Substituting ,Tows. This point solves , (6fuv into (10), we obtain the solution of (5), tionary point of functional (4). Concerning the synthesis problems of linwhich is a staear radiator for the case of one-dimensional domains  the solu-tions of system of equations similar to (13)re investi-gated, in particular, in . The obtained there results show that nonuniquness and branching of solutions, de-pendent on the size of physical parameters of the prob-lem are characteristic for special case of equations of type (13) (when variables are separated). The results  can not be transferred directly to two-dimensional nonlinear integral equations of type (7). Here unlike the branching points  there exist branching lines of solu-tions, and the problem of finding the branching lines is not enough investigated nonlinear two-parameter spectral problem. a. Equations of a Set of Branching Points the case when the kernel of (7) determined by (8) is 3 Inreal, (7) in a space of real continuous functions C has the form  ,, d,,sign d.KQQfQ QfQFQKQQfQ Qcc (23) Assuming in (23) sign 1fQ tegral linear we obtain the sec-ond kind Fredholm inequation with sym-metric and even kernel  1,, dcfQKQQ fQQQF. (24) Here the right part 1QF,,dcFQKQQQ is a nonnegative function. It was shown that correspond-ing to (24) homogeneous Equation (18) has only zero solution. From here follows that (24) has unique solution 0()fQ, belonging to the space ()C and 0( )1Qsign f, that is solution 0()fQ is nonnegative. Further welution l shall call the so0(,)fQc as initiaso tion lution of (7). Corresponding to it solu00,,,ccuQ fQ0,0cvQ we shall call as initial solution of a system of Equa-tions (13). To find the branching lines and complex solutions of (7), which branch-off from the real solution 0,cfQ we consider the problem on finding such a sees t of valuof parameters 00012,ccc and all distinct from ,cfQ solutioh satisfy the conditions ns of (13), whic00max ,,0,ccuQfQmax ,0,cQQvQ (25) Copyright © 2011 SciRes. AM P. SAVENKO ET AL.1081 as 00cc. Conditions (25) mean that it is necessary to find small continuous solutions in  0,, ,cc cwQuQf Q , 0,,ccQvQ, converging uniformly to zero as cis necessary also to take into accou0cnt th. In addition, it e direction of convergence of vector toPut c 0c. 00112 2, ccc c  (26) and we shall find the deired sotioslun in the form 00,, ,, ,,,. (27) vQ QcuQf QwQccFurther we omit the dependence of functions ,,wQ and ,,Q on parameters  and  for reduction of notations. Present some properties of integrands in (13). They are coand (27) it i-r series with respect to the funtinuous functions of arguments. We substitute (26) n (13). Then inegrands expand in the unformly convergence powenctional arguments w,  and numerical parameters ,  in the neighborhood of point 000,,,0ccfQ :   2200,, uQKQQuQ Q c,, ,mnpqmnpqmn pqFQ vQAQQwQ Qc u 2201,,,, .mnpmnpqmn pqvQKQQvQ FQuQ vQBQQwQ Qq cc(28) Here ,  are cficientsi on argu-ments. Su , we obtain a 0,,cmnpqAQQ of expansion contbstituting (28) in 0 as a soluf nonlinear equation0,,cmnpqBQQnuously dependent(13) and consideringtion of this systems with respect to oef-0stem osmall solu-tions w and ,cfQsy:   000021,,d,,, d,pqm nmnpqmn pqQQFQKQQQfQBQQwQ QQ  ccc (30) where 0010 00100001 0001,,,,,,aQA QQQaQAQQQcccc d,d.Extracting in a system of Equations (29), (30) the lin-ear members for and w, to find a set of brancf the sohing  000,, d,10 0102,2π,, d,  pqm nmnpqmn pqwQa aQK QQwQQcQAQQw QQQcc (29) ccpoints olution 0c al equtions,fQawe have an integf linear inr rated system oteg1,, dcwQK QQwQQ  , (31) 100,,,d.,   QQFQfQ KQQQfQccc (32) We shall call a set of values of parameter  00012,cccneous Equationstions as a set ofsolution , at which the system of linear (31), (32) has distinct from zero points of possible branching of initial homoge- solu-0,cfQ. Note that the system (31), (32) is ely the functions and noncoherent relativ w. Prevhown that equatiof ty31) has i-ously it has been son pe (only a zero solution. Therefore the premndiobl of fing a set of branching points under condition0fQ,c0, is reduced to equation  100,,, d,, QTQFQfQ KQQQfQcccc (33) that is to a nonlinear two-dimensional spectral problem [14,15]. The eigenvalues of this equatio of n form a setpoints of possible branching of solutions of a system of nonlinear Equations (29), (30). Corresponding eigen-functions are used to construct the branching-off sol-utions of equations . Note, that construction and justification of conver-gence of numerical algorithms to solve the nonlinear two-parametric spectral problem on eigenvalues (33) it is necessary to solve the corresponding auxiliary one-pa- rameter nonlinear spectral problem . In this connec- tion at first we consider two-parameter nonlinear spectral problem on the general (operational) level in the Banach spaces. Copyright © 2011 SciRes. AM P. SAVENKO ET AL. 1082 Nonlinear Spectral 4. Two-DimensionalProblem 4.1. Statement of the Problem. Existence Conditions of Descrete Spectrum of Operator-Function Note that different approaches are used to discretization of the original problems [7,14] at construction of the numerical algorithms to find the solutions of various typr spectral problems. That is to say that es of nonlineaoriginal problems in the Banach functional spaces E are replaced by the corresponding problems in the fi-nite- he quesmate solutions of iscretization problems totions of initial prob-dimensional spaces nE (n). In addition ttion of convergence of ex approxiact soludlems, whenever dim nE, n is important, since the input and approximated equations are consid-ered in various spaces. Let E be a complex Banach space, 12, be a vector parameter belonging to the domain 12  (open connected set) of the complex space 2. Here ii , :iiiir 1, 2i, r be some real con- stant. Consider the ctioperator-funon ,: ,EE AL, which to each 12, as thsignse operator 12,,EEAL, where ,EEL is the space of bounded linear operators . coproblem of Let usnsider the nonlinear two-parameter spectral the form12,0xA, (34) in which it is necessary to find the eigenvalues 00,12 to them eigen-and corresponding vectors (0)xE(0)(x0012,0) such that 00xA. In pan rticular, in view of (33), the operator-functio12, is represented as 12,TAA12,:IL,EE. (35) Here 12,T is a linear completely continerator actiuous op- the Bang innach space EC and ana-t on two-dimensional parameter 12,lyticallyn depende, I is a un Banach sique . paces and also thoperator in Let the E, EEnne system ()nnpP of linear bounded operators nE such that 1, 2,:npE,,nnE Epxxnx E (36) be. Operato bo givenrs np are called conjunctive opera-tors [7,14]. From the principle of uniformundedness  for p the inequality follows nconst npn. very space E the elemLet in ennxent be selected. Writing these elements in order to increase the numbers we shall form the sequence nx. her approachUsing eit scretizaton of oriproblem the operator-functionto diiginal ,EE is ,: ALapproximated, respectly, by the approximate opera-tor-functions ive,: ,nnEE nAL, n. As a re-sult, at each 12, we obtain a sequence of operators (,)nnnEEAL which convergences to opera-tor (, )EEAL at satisfaction of theorresponding conditions. Definition of various type convergence of opera-tors cs ofm (33)ifIn particular, described by (3 discnA to A is given, in particular, in . Discretization of original proble, choce o the spaces nE and determination of operators :nnpE E are realized in various ways. the opera-tor-function is5) and E is the separable (infinite-dimensional) Hilbertian space, one of the ap-proaches toretization (34) consists in the following. Consider an arbitrary complete orthonormal in E sys-tem if of functions 1kkx. Each element xE can be presened as series t 1kkkxcx where ,xxkkthe Four coefficient of element c is riex. If T12, is ata linear continuous operator, acting in the separable Hil-bertian space it admits matrix represention :  1212 ,1,,MjkjkTt , (37) where 12 12,,,jkkjtTxx. In addition a se- quence of the Fourier coefficients lement of e12,yTx is obtaia sequencerier ned from of the Foucoefficients of element x as a result of multiplication of the matrix 12(,)MT by coefficients ot f elemenx. Using the matrix representation of operator 12,MT the spectral problem (34) is formulated as 12 12,,MMMxT Ix 0AIM is a unit matrix in the s, (38) where pace of sequences 2l.Thus, the operators ()T and )MT( are equiva-lent in the sense that they to the same element xE assign one and the same element yE. But we obtain the Fourier coefficients of element yx as a result of action of or T on the element the operatMx. Os coincide,bviously, that eigenvalues of these operator that is the spectral problems (34) and (38) are equivalent. In this case we put that the finite dimensional spaces nE are generated by the bases 1kkx (n) and to each nelement xE the operators :nnpE E assign the element 1nkkkxcx where ,kkcxx. As a , operator resultnMT approximated to s described MT iCopyright © 2011 SciRes. AM P. SAVENKO ET AL.1083 rix-fuby the finite-dimensional matnction  1212 ,1,,nnMjkjkTt. (39) Apply other methods of discretization to (34), in par-ticular, the quadrature (cubproce for the case ature) ssesof homogs integral equationge of deri-eneouns and chavates by tnce analogfferential equa-heir differeues in ditions. Weroximate o find approxi- obtain appproblems tmate the lues and eigenvectorsmatrix opera-eigenva of tor-functions in the form 12,0, nnxnA. Moreover, the problem of determination the eigenval-ue(41) If (40) s is reduced to finding the roots of the determinant of n-th order that is the roots of equation   12111 2121 211 2211 22212212,,, ,,, ,det   nnnaa aaa a 0,, ,   1122 1212 nn nnaa an12,nA has form of (35), then ,.kkNote if the coefficients  ,12,12,12,,,,1,jkjkjktjatj  12,ija arguments, the are continuously differentiable functions of partial deriva-tives 12,nj (j1, 2) vation . are determined by the rules of determinant derider the auxiliary eter stral problem, necial case of (34). Assume that variable Consi nonlinear one-paramcessary later on, as a spepec-2 in the operator-function 12,A ated function is expressed ome one-valued fferenti-by sdi21z mapping the subdomain omain 1, 1 into some subd2,2. In the simplest case we put 21 ( is some real pa-rametenoperatunction 111,zr). Itroduce the or-fAA for 11,, which is reduction of the operator-function 12,A. We shall consider one-dimensional spectral problem 10x (42) Ain which we assign to each value 11,z the operator 11,,zEEAL. Analogously to (40) we consider a approximate sequence of discrete problem of (42) 1,0 z n (43) Denote the spectrum of operator-function ,1 ,nnxA1A as sA. Assume that 1,sA. For the spectrum sAof the probwlem (34) the fol-lowing theorem is valid. Theorem 2. Let the folloing conditions be satisfied: 1) operator-function ,: ,EE AL is holo-morphic, and sA; 2) operator-functions ,:EE AL,ny closed bounded snnet 0 are holomorphic and for an the following inequality 12x ,nc0maAconst ()n is valid; 3) operators  12,,EEAL, ,,nEEAL ()12 nnn are the Fredholm op-erators with zero index for any 12,; 4) spectrum 1,sA of func-tions and a sequence12n, are differentiable in the domain ; 5) nAA is stable for any \rsAA. Then the following statements are true: 1) every point of spectrum 01sA is isolated, it is perator eigenvalue of the o111,z,AA the finite-dimensional eigubspa censce 01NA and the finite-dimensional root subspace t; orrespond to i2) for each 01sA there exists a sequence 01,n,nsA 0()nn, s0n1, from uch that 001, 1n; 3) each point  000,11z is a spectrum int of the operator-function12,A; po4) if in some 0- neighborhood of the point  00011,z 122,0nz , then in anll arbitrarily sma-neighexists a contiuous differenti unctiborhood of that point there nable fon 2, 1, 1), that is ibicylindric dwhich is solution of (4n some omain al 0012011 2,:012 2,  th m of the operator-function ere exists a connected component of spectru2,N (1,21A are small real constants). plete spectrProof. The proof of Theorem is based on Theorems 1, 2 [14, p. 68, 69] and on existence of implicit functions (see, exam, ). At first we show that the conditions of Theorem 1 [14, p. 68] concerning the existence of discreum of operator-function 1Arem. follow from the conditions of formulated Theo Under the conditions of Theorem the operator 12,A is Fred-holm operator with zero index for each 12,, and the operator-fu ,: ,EE is holo- mo Fronction ALrphic. m here follows that at each 11, the operator 1A, as reduction of the operator 12,A, is also Fredholm operator with zero index, and the op-Copyright © 2011 SciRes. AM P. SAVENKO ET AL. 1084 erator-function  11,:,EEAL is holomorphic. So, for the operator-function 1A the conditions of Theorem 1 [14, p. 68] are satisfied, from which follows: each point 01()sA is isolated, it is the eigenvalue of the operator 1()A, the finite-dal eigen-subspace and finite root subspace corres, each point 0011,z is thespectrum point of thtor-function imp0raensionond to it. Thus e ope12,A. In addition, the conditions of Theorem 2 [14, p. 69] are satisfied for the operator-function 11,:,EEAL. From this t follows: at n larger than some 0n for each heorem01sA there exists a sequence1,n  from 1, ,nnsA such that 01, 1n. Thus, each point (0)(0)12,  0011,z is the eigenvalue of operator-function 11,:Atively, the eigen-,EEL and, respecvalue of operator-function ,:  nnence,EE .n. AL0 Since  (44) 1, ,nnsAis the root of (41), then from here follows that  001, 1,,0, nzFrom the converg of sequence 0sA to 0s1, ,nn1A follows arbitr that for anarily small num-ber 0 there existsber such num0Nn that  001, N1001, 1,,0NNNz and . Let 1, 2 be independent variables in the domain , and 0 0121 1zbe a spectrum operator-function 00,,  point of the12,A belonging to ssAAfunctions . Under the conditions of Theorem the 12,n are differentiablezbo in and the neigh- rhood of the point 00012 1, 01,0N01,,0Nz. In addition tt 1,2Nh0gs to e poin01, 1,,NNz belon-vicinity o. According to the Theorem ab of poi1f thet ere ex poin0012,out implicit 00function in some neighborhoodnt 12,th ists the continuous differentiable function 2N quation 12,0N, and 1, 1,nNztenc, solving the e00N. From here fopectrum component of thellows exise of connected sor-func- tion operat,:A,EEL in some bicylindrical domain 00012,0111222: , , where 1ε, 2 are small real constants. oved. Theorem is prComment. Note that in the case of real para1meters  and 2 the presence of a singular point in the equation 12,0  is one of the sus fficient criterionof action of condition sA. The point satisf0012, is a singular point of the curve which is prese,0nted by the equation 12 when   00121,0, 0012,d the second 20, anorre nonzero: der partial derivatives a 0020021212 ,,0, 0,221200212,120().  These derivatives and tr derivatihe third-ordeves are continuous in the neighborhood of the point 0012,. If in addition    200002212 12,,  002122212 12,0  , then the point 0012, is the second ordeot of r roequation 12(,) 0. Inside of a sufficiently small ra00dius circle with center at point 12, the left part of equation 12(,) 0 becomes zero only at point (0)(0)12(,) (0) (0)12(,) is the i, i.e. solated point of spectrum. 4.2. Finding the Connected Components of a Spectrum the exiThus, assumingstence of discrete spectrum 1,sA robland solviem (42), set of the eigenvng an auxiliary one-dimensional spectral pwe find a alues  000,11z, whichalso to the belongs perator-funcspectrum of otion ,: ,EE AL. To find the ccted comonneponents of a spectrum in some neighborhoods of the points  000 11,z consider thewe problem on finding the solutions of quation e12,0n, as the problem on finding the implicitly given function 221 at satisfaction of  condition 0012 2,0n  (or 112 at  00satisfaction of condition 121,0n ), solv- ing the corresponding Cauchy problem   0012 12,dn002,112dn , (45) Copyright © 2011 SciRes. AM P. SAVENKO ET AL.1085 ne-diorm 0n. (47) alues of this problem n. (48) Solving the problem (45), (46) in some neigof the point co001 1z . (46) Corresponding (38) auxiliary omensional spectral s the f2problem hannObviously, the eigenvare the 11 11,,nnMMMzxT zIxAroots of equation  111det,0nnMM MTz I hborhood  12,iiiz, we find the i-th nnected component of spectrum of the operator-func- tion 12,nMA. Return to finding the solutions of (33), in which are real spectral parameters. Let c, 12c12,c, cc12ccc , where :0iicicicccr . By direct check we set that for arbitrary values of parameters 12,ccc  the funcisnctions, where f (7). Write the necestion  ,,ccQfQ, (00 one of the eigenfu is the ini-49) ,cfQsa0tial solution ory in what follows equation   0,,,, Q0dQTFQfQKQQfQccQcc0 (5 ) conjugate with (33). For arbitrary 12,ccc  the function 0 0,,ccQFQfQ (51) is one of the eigenfunctions of (33). The existence of distinct from identical to zero solu-tions of (33) for arbitrary ,cc  indicates that there i12 cnt of spectrhe condition . To satisfy th, (51) from ths a connected componeum, coinciding with the domain. So, tof Theorem 2: is nois conditgenfunct (49)e kernel of in-on ct satisfiedionscsA exclude eiion we tegral equation (33). Consider the equati ,,,cccQT QQ KdQQ , (52) where  000000,, ,,,,,QQK QQfQQQccccK. (53) From the Schmidt Lemma [16, p. 132] follows that 1,FQ fQcc is not characteristic value of (52) for any value ,cc , that is  is not an eigenfun12equation. 0,cQeby the connen c cluded fromction of this Ther cted component domaiand corresponding to the function is ex the spectrum of operator. coinciding with th0,cQe Using (8) we are sure that for the kernel of operator cT is fulfilled the inequality  031222 20221202d,2π,d .2πQQcc QQ222,, dd,1QQQQQcccc  ccKHere  is the measure of the domain . From e obtained inequality follows that cT is the Fred-thholm. Moreover, it is a operator with zero index completely continuous operator in space 2L . Functions entering in the kernel of thr (35), e operatoadmit the analytic continuation into the complex domain c, if and are assumed as complex parameters. 1c2c Holomhy ofrator-function orp ope12,cc A ,c12Tc I llows  from existence of partial foderies vativ12,ccAic 1, 2i at arbitrary pot in0012,ccc  owing to continuity ofkernel the ,,cQQK according to a set of their variables in the domain c and existence and continuity of partial drivative es12,icccK 1, 2i, what is easy to verify. 52), 21cc, we shall consider the onPutting in (e- dimensional o , (54) spectral prblem  11,, dQTcuQQc QQKGwhere is ,, ,,,QQ cQQccKK. Since Tc111 1f the operator ()Tc, from here follows that reduction oTc is tholm operator with zero index for ane Fredhy 111,c, and the operator-function 1:,IEEL is holomorphic. TAIf 1sA, from holomorphy of operator-function and from the Fredholm property of the kernel 111,, ,,,QQ cQQccKK satisfaction of the con- ditions of Theorem 2 follows. In accordance with this Theorem every point 0cs is i it is1solated and the Aeigenv00alue of (54). Respectively, the points 12,cc  01c are eigenvalues of (5401,c). To find the spec-trum connected components (s of 52) in the vicinitiepoints (0) (0)12,cc we solve the Cauchy problem (45) Copyright © 2011 SciRes. AM P. SAVENKO ET AL. 1086 an46), usingd ( the found solutions of auxiliary problem (54) as initial conditions. ically the ei-iary problem(43). Consider4.3. Numerical Finding the Eigenvalues of the Problem We shall construct algorithms to find numergenvalues of (54) what corresponds to auxil some convergent cubature process  1d,njn jnnjxQQa xQxn (55) with coefficients jna and nodes jnQ (1jn). We reject the remainder term nx in (55) nd replace integral in (52a) by it. Giving the variable in thsional spaces Q us values inQQ (1in), we have the homogeneosystem of linear algebraic equations concerning ,u: 1,nnnu111,,, 1nninMnjnin jnjnjuTcuaQQcuin K(56) where uuQ. Solving the eigenvalue problem (56)in ine finite-dimennEC,n wapproxialues, convergent to exact solutions as Finding , inThese stions of equation we denote as . Return to two-dimensional spectral probplying the (55) to (52), we obtain a system of linns similar to,0n (59) we consider as a problem on finding the implfu chylem (4e find mate eigenvof the problem (54). the eigenvalues of (56) is reduced particu-lar, to finding the roots of equation  11det 0nnMncTcI. (57) olun1iclem (52). Ap-ear equatio (56) 12 121,, nninMjnin jnjnjuTcc uaQuK (58) , 1.Q ccinFinding the solutions of equations  12 12,det ,nnMccT ccI  icitly given nction 221ccc, reducing it to the Cau prob5), (46). Since to each isolated root of this equation corresponds eigenvalue  nditions 121 1,,iii iccc c of prob- lem (58) we use solutions of (57) as the initial co(46). Thus, we determine the initial conditions (46) for the Cauchy problem as . If  211iiiccc12 2,0ncc c then solving the problem (45) (46) in the differentiable each vicinity of points 1ic, we findfunction 21icc which satisfies the condition 1iiccIn the case, when 11,iiccis of a real eigenvaluethe problem (57), 21icc are real differentiable functions describing in the vicinity of points  11,iicc some smooth curves. Tse thhat is, in this cae equations (52) and (33) have a linear spectrum, respectively. Thus, solving the proble (57) and (58), we find a setms of values of parameters c at which the bran-12 c,c ching of ctions of Equation (7) from the real omplex soluinitial solution 0f,cQ at 0, 0 is possible. U Functional has smon branching- aller values ofhe solution xamplesf solutions than on t,cfQ . 5. Algorithm of Finding the Solutions of Nonlinear Equation. Numerical E 0 Present one of iterative processes for numerical finding the solutions of system (13), based on the successive approximations method: 11112122,, 0,1,.nnnvIBBuvn 11111121,,nnnuIBBuv(60) Before it was shown that the inverse operators 11. 111IBited.  and 21IBexist and are lim11 In the case of even on both arguments function ,F12ss and symmetric domains G and  at exe-cution of iterative process (60) it is appropriate to use the invariance property of the integral operators B1(u,v) and B2(u,v) in the system (13) concerning the type of parity of functions u(s1,s2), v(s1,s2). Functions and hav-ing some type of parity on the correspng ambelong to the invariant sets and of uondiklV vrguthe sent pace ijU()C where the indices taes 0 1. In pa,,,klij ke valu orrticular, if 12 01,us sU then 12,uss , 12,uss12,us s. By the 1s2,us direct check wone are ct there are such inclu-vinced thasions: 12,,Bij klijijklklij klij klBUVUBUVVUV UV (61) From these relations, in particular, follows the possi-bility of existence of fixed points of operator B be-longing to the corresponding invariant set, that is to solu-tions of (13) and, respectively,). (7Substitute into (10) the function arg nfQ arctg nnvQuQ f succes- obtained on the basis osive approximations (60). As a result we have a sequence of function values which we denote as {Un}. For this sequence the Theorem 4.3.2 and corollary 4.3.1 , and Copyright © 2011 SciRes. AM P. SAVENKO ET AL.1087 the Theorem 4 from  are valid. From here follows that the sequence {Un} is relaxation for functional (7) and the numerical sequence nU is convergent. Consider the numerical examples of approximation of even on both arguments function  121 2,sinπssinπsFss  (Figure 1) in the domain 12 12,: 1,1ssss . In Figure 2 in logarithmic scale the values of func-tional U, which it takes on different types of solu-tions of system of (13) at change of the parameters 12,cc on the beams 210.8cc are presented. The curve 1 corresponds to the initial solution 012,fss . The curve 2 is a branching-off solution at point  o11, 2.345,1.876cc with the prrty 12 pe 12 1arg ,arg ,2fss fsshe analysis of . From t Figure 1. The function  1212,sinπssinπsFss  given in the domain :12 12,1,ssss.1 FFigure follows that branching-off solutions at point  1112,cc are more effective compared with initial solu- igure 2. The values of functional on initial and branching- off solutions. tion , since the functional 0fU accepts smaller valu branching-off solution initial. es onns than oThe points of possible branching of solutions of (7) (spectral lines of (33)) for given 1212,sinπssinπsFss are shown in Figure 3. 12,fssFigures 4 and 5 present and 12arg ,fss of approximate function corresponding to the branch-ing-off solution of system of Equations (13) at с1 = 8 and с2 = 6.4. Correspoding to this solution the functions ,Uxy and arg ,Uxy of the Fourier integral prototype in a spatial image and image by the level lines are shown, respectively, in Figures 6 and 7. Figure 3. Points of possible branching of solutions (spectral lines) for given 1212,sinπssinπsFss . Figure 4. The modulus of approximation function. Copyright © 2011 SciRes. AM P. SAVENKO ET AL. 1088 Figure 5. The argument of approximation function. Figure 6. The amplitude of Fouier integral prototype. Figure 7. The argument of Fouier integral prototype. ,Uxye domainAs we see in these figures, the function is a nonsymmetric relatively to the center of th along the axis , and accepts the 0 or in thondins of the dom, t is functG value ain OXe correspion arg ,Uxying subdomaπ hatG,Uxy is real. that the Thus, fromgiven symmefunction the tric analysis of Figures follows12,Fss (even on both arguments) on the beam 210.8cc at is approximated effec-tively btimce of nonsymmetrical by modulntegypes, that is the functions 111ccal choiral prototy the opus Fourier i,.y Ux 6. To Selection of Parameter  We shall present some argumentations concerning the choice of the weight parameter of regularization  in the optimization criteria (4). Many of works (see, [23-26 ])areof soly for l devoted to this question. Efficient algorithms lving this problem are developed maininear operator equations of the type LuF (62) with approximate right part F, in which a priori the ror er is known. Thiled definition is given in eir deta[23,25]. The principle of residual is the most applicable one in practice. Here we choose such number  for which acy the equality with the necessary accurLuF (63) is execu is a minimum point of smoothing func-uted (tional defid on UHne ) dependent on the parameter . As shown in , the principle of residual (63) to deter-mine the parameter  in the case of nonlinear operator L can be applied when L is a convex operator. Gen-erally, the residual (63) may be discontinuous or on-nmonotonous function with respect to parameter . Therefore Equation (62) can not have any solution or have a set of solutions. The error  is unknown a priori, as a rule, in the problems of nonlinear approximation. Decrease of pa-rameter  in the functional (U) reduces the require-σαments to the norm U. As a result, the norm of the Fourier integral prototype minimizing the functional σα(U), inversely depend on . At reduction  the accuracy of approximation in the limits of the dain om, as rule, increases, but the value of function 12,fss outside this domain increases also. At concrete calculating the parameter  can be se-lected on the basis of some physical argumentations ad numerical experiments. In particular, in the antenna syn-thesis problems the rameter npa can be selected from e satisfaction of equal engy condition  ther 2222UF. (64) LLG Copyright © 2011 SciRes. AM P. SAVENKO ET AL.1089 (A numerical example of dependence of solutions of7) on the value of parameter  at approximation of the function 112 2cossinπ2πs,Fss s is the proof of the above presented arguments. In Figure 8 are given the values oapproximate function f 12,fss in the sec-tion 10s. From the analysis of the Figure, we see that the quality of approximation of the given function on the interval 211s  increases hen parameter w de- creases, while 20,fs increases outside this interval. 7.the modulus of double Fourier transformsmi method of solving the twnlineari sase of even by both arguments (one argument) function Conclusions Note the main results and problems arising at investiga-tions of the considered class of problems: 1) The method of nonlinear approximation of finite nonnegative functions with respect to two variables by with use the oothing functionals is developed in the work. 2) It is shown that non-unqueness and branching of solutions is characteristic for this class of problems. The numericalo-parametric nonlin-ear spectral problem enabling to find the branching lines of solutions of Hammerstein type no two-dimen- sional integral Equation (7) is proposed to study the non-uniqueness of solutions dependent on the value of parameters ,cc entering the Fourier integral. 3) At fsolutions of system of Equation (13) 12nding theby the successive approximations method (60) in the c12,Fss to obtainecessary to select th the solution of concrete type it is e initial approximation belonging to nthe corresponding invariant set of nonlinear operators Figure 8. The modulus of approximate function 2112π,cossinπ2sFss s in the section 20s correspond- ing to various regularization parameters . 1B, 4)2B Inve (61). stigations of branching of existing solutions de- pendent on physical parameters entering the Fou-rier integral are the main difficulving this class of problems. As follows from the ted researches, in particular, in [4,13], for a spec12,cclty at sopresenial case when 121 122,FssFsF stions with increase of pa, the f existing solu- significantly quantity orameters 12,ccincreases. Note, that in many practical applications, par-ticularly in the synthesis probleing systems, ms of radiatobtaining the best approximativen function on to the gi12,Fss at concrete valuesters is of parame12,ccimportant. It allows to limit onestig of eself to invationsfew first points (lines) of branching. 5) Obtaining the complete answer about exact qutity of thpa-dies. .11008ane of existing solutions of (7) at concrete values rameters ,cc are the subject of separate stu128. References A. N. Tikhonov and V. Y. Arsenin, “The Methods of Solution of Incorrect Problems,” Nauka, Moscow, 1979.  P. Savenko and M. Tkach, “Numerical Approximation of Real Finite Nonnegative Function by the Modulus of De-screte Fourier Transform,” Applied Mathematics, Vol. 1, No. 1, 2010, pp. 65-75. doi:10.4236/am.2010  E. G. Zelkin and S. I. s andInstid Ma, “Elements of Func-. VaiBran- of Solutions of . in, “The Functional Analysis,” Nauka, Solov’yev, “Methods of Synthesis of Antennas: Phased Antenna Array Antennas with Continuous Aperture,” Sovet radio, Moscow, 1980.  P. O. Savenko, “Nonlinear Problems of Radiating Sys- tems Synthesis (Theory and Methods of the Solution),” tute for Applied Problems in Mechanics anthe-matics, Lviv, 2002.  A. N. Kolmogorov and S. V. Fomintions Theory and Functional Analysis,” Nauka, Moscow, 1968.  М. Mnberg and V. А. Trenogin, “Theory of ching Nonlinear Equations,” Nauka,Moscow, 1969] V. A. Trenog[7Moscow, 1980.  М. А. Krasnoselskii, G. М. Vainikko and P. P. Zabreiko, “Approximate Solution of Operational Equations,” Nauka, Moscow, 1969.  L. V. Kantorovich and G. P. 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