 Applied Mathematics, 2010, 1, 65-75 doi:10.4236/am.2010.11008 Published Online May 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform Petro Savenko, Myroslava Tkach Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine Email: {spo, tmd}@iapmm.lviv.ua Received March 15, 2010; revised April 21, 2010; accepted April 28, 2010 Abstract The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear prob-lem on mean-square approximation of a real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on two parameters, are constructed and justified. Keywords: Mean-Square Approximation, Discrete Fourier Transform, Two-Dimensional Nonlinear Integral Equation, Nonuniqueness And Branching of Solutions. 1. Introduction The mean-square approximation of real finite nonnega-tive function with respect to two variables by the mod-ulus of double discrete Fourier transform dependent on physical parameters, is widely used, in particular, at modeling and solution of the synthesis problems of dif-ferent types of antenna arrays, signal processing etc. [1-3]. Nonuniqueness and branching of solutions are essential features of nonlinear approximation problem which remains unexplored. The problem on finding the set of branching points, in turn, is not adequately ex-plored nonlinear spectral two-parametric problem. The methods of investigation and numerical finding the solu-tions of one-parametric spectral problems at presence of discrete spectrum [4-8] are most well-developed. The existence of coherent components of spectrum, which are spectral lines for the case of real parameters , is essen-tial difference of nonlinear two-parametric spectral problems. In the work a variational problem on the best mean- square approximation of a real finite nonnegative func-tion by the module of double discrete Fourier transform is reduced to finding the solutions of Hammerstein type nonlinear two-dimensional integral equation. Using the Schauder principle the existence of solutions is proved. The existence theorem of coherent components of spec-trum of holomorphic matrix functions dependent on two spectral parameters is proved. It justifies the application of implicit functions methods to multiparametric spectral problems . The applicability of this theorem to the analysis of spectrum of two-dimensional integral homo-geneous equation to which is reduced the problem on finding the lines of possible branching of solutions of the Hammerstein equation, is shown. Algorithms for numer-ical finding the optimum solutions of an approximation problem are constructed and justified. Numerical exam-ples are presented. 2. Problem Formulation, Basic Equations and Relations Consider the special case of double discrete Fourier transform ( )2211()121122()(,)expNMnnmnmnmnNmMnfssIicxscys=−=−=+∑∑ %% setting here nmxxn=∆ ()nNN=−÷ , nmyym=∆ ()mMM=−÷ ; 11xcc=∆%, 22ycc=∆%. If it is necessary for the accepted assumptions we shall consider the for-mula ( )121122(,)expNMnmnNmMfssAIicnscms=−=−=≡+∑∑I (1) as a linear operator, acting from complex finite-dimensional space 22NMIH×=£ (221NN=+, 221MM=+) into the space of complex-valued continuous functions with respect to two real variables determined in the domain (){}121122,:,ssscscΩ=≤π≤π . P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 66 Here 1c, 2c are any real non-dimensional numerical parameters belonging to (){}1212,:0,0ccccacbΛ=<≤<≤. The function 12(,)fss is 12cπ - periodic function on argument 1s and 22cπ - periodic on 2s. In considered spaces we introduce scalar products and generable by them norms ( )212 124,INMnmnmHnNmMIIcc =−=−π=∑∑II , ()1/2,IH=III, (2) ()( )()()(2)1211221212,,,CfffssfssdsdsΩΩ=∫∫ , ()( )(2)1/2,CfffΩ=. (3) Denote an augmented space of continuous functions with entered scalar product and norm (3) as (2)()CΩ and notice that its augmentation coincides with the Hilbert space 2()LΩ . By direct check we are sure that such equality ( )2222 1212 ,,nmnmAfssdsdsIIIΩ===∑∫∫ (4) is valid. From here follows, that A is isometric opera-tor in sense (4). Using the entered scalar products (2) and (3) we find the conjugate operator required later on ( )()12121122122,exp4ccAffssicnscmsdsds∗Ω=−+π∫∫ (),nNNmMM=−÷=−÷ . (5) Let such function be given 12121212(,),(,),(,)0,(,)\,FssssGFssssG∈⊆Ω=∈Ω% (6) where 12(,)Fss is a real continuous and nonnegative in the domain G function. Consider a problem on the best mean-square approxi-mation of the function 12(,)Fss in the domain Ω by the module of double discrete Fourier transform (1) ow-ing to select of coefficients of the vector I. We shall formulate it as a minimization problem of the functional ()( )(2)(2)22() FAFfCCIIΩΩσ=−≡− (7) in the Hilbertian space IH. Taking into account (4) and (5), we write the functional ()Iσ in a simplified form ()( )()(2) (2)222()2,IHFFACCIIIΩΩσ=−+. (8) On the basis of necessary condition of functional minimum we obtain a nonlinear system of equations re-lating to the components of vector I in the space IH that are represented in the vector and expanded forms, respectively: ( ){}expargAFiAII∗=, (9) ( )12 122,exparg4NMnmnmkNlMccIFssiI=−=−Ω=× π∑∑∫∫( ))( )}1122112212expicksclscnscmsdsds×+−+  (),nNNmMM=−÷=−÷ . (9′) Acting on both parts of (9) by operator A we obtain equivalent to (9) the Hammerstein type nonlinear integral equation relating to f: ()[](),,()exparg()fQfKQQFQifQdQBcΩ′′′′=≡∫∫ , (10) where 12(,)Qss′′′=,12dQdsds′′′=,()12,ccc=; ()()()11112222,,,,,,KQQKsscKsscc′′′=⋅ , (11) ()( )11111111,,exp2NnNcKsscicnss=−′′=−≡π∑ ( )( )21 111111sin 222Ncssccss′−≡π′−, ()( )22222222,,exp2MmMcKsscicmss=−′′=−≡π∑ ( )( )22 222222sin 222Mc ssccss′−≡π′−. Note, that the kernel (11) of Equation (10) is degene-rate and real. We shall consider one of the properties of function ()exparg()ifQ′ entering into (10) at ()0fQ′→. Ob-viously that the function ( )()1222()()()exparg() ()()()fQuQivQifQ fQ uQvQ′′′+′=≡′′′+ is continuous if ()Re()uQfQ′′= and ()Im()vQfQ′′= ()Im()vQfQ′′= are continuous functions, where (exparg()1ifQ )exparg()1ifQ′= for any ()fQ′. If ()0uQ′→ and ()0vQ′→ simultaneously then ()0fQ′→ is a complex zero. Its argument is undetermined accordingly to definition of P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 67complex zero [11, p. 20]. On this basis we redefine ()exparg()ifQ′ at ()0uQ′→ and ()0vQ′→ as a function which has module equal to unit and undeter-mined argument. The equivalence of (9) and (10) follows from the fol-lowing lemma. Lemma 1. Between solutions of Equations (9) and (10) there exists bijection, i.e., if I∗ is a solution of (9) then fAI∗∗= is the solution of (10); on the contrary, if f∗ is the solution of (10) then []{}exparg()AFifI∗∗∗= (12) is the solution of (9). Proof. Let I∗ be a solution of (9). Then AFiA∗∗∗−≡II ( ){}exparg0AFiA∗∗∗−≡II. Acting on this identity by the linear operator A, we have ({exparg0AAAFiAII∗∗∗−≡ ()}exparg0AAAFiAII∗∗−≡. Since the operator A acts from the space 22NMIH×=£ into the space (2)()CΩ and accordingly into the space 2()fHL=Ω%, and the set of its nulls consists of only null element from the last identity follows, what fAfHI∗∗=∈% is a solution of (10). On the contrary, let ffH∗∈% solves the Equation (10). The operator A∗ acts from the space 2()fHL∗∗=Ω% into the space 22NMIH×=£  and the Hilbertian space 2L∗ coincides with the space 2L . From here follows, that A∗ acts from the space 2()fHL=Ω% into the space 22NMIH×=£. Taking into account that F is a finite function determined by (6), and f∗ is conti-nuous, the function ()exparg()Fif∗ is quadratic inte-grability in the domain Ω, i.e. ()exparg()fFifH∗∈. Thus ()()exparg()IAFifH∗∗∗=∈I and the right part of (10) is the result of action of operator A on an ele-ment I∗, i.e. ()()exparg()AAAFiffI∗∗∗∗==. Writ-ing this equality as ( )()()exparg()0AAFiAII∗∗∗−= and taking into account that a set of operator nulls con-sists of only a null element we obtain ()()exparg()AFiAII∗∗∗=. So, ((exparg()AFifI∗∗∗= ))exparg()AFif∗∗ solves the Equation (9). Lemma is proved. Thus owing to the equivalence of (9) and (10) we con-sider simpler of them, namely (10). The Equation (9) is a more complicated equation in sense that in its right part the operator A is in an index of the power of exponent. Besides taking into account that a set of values of opera-tor A is a set of continuous functions in the domain Ω belonging to the space ()2LΩ and this set is a compact in the space 2()LΩ , we shall investigate solutions of (10) in the space ()CΩ. Formulate the important properties of (10), which are checked directly. 1) If function ()fQ is a solution of (10) then the conjugate complex function ()fQ is also the solution of (10). 2) If function ()fQ is a solution of (10), then ()exp()ifQβ is also the solution of (10) (β is any real constant). 3) For even on two arguments (or on one argument) functions 12(,)Fss the nonlinear operator B that is in the right part of (10), is an invariant concerning the type of parity of the function 12arg(,)fss on two arguments (or on that argument on which 12(,)Fss is an even function). Below taking into account the property 2) for unique-ness of solutions we set the parameter 0β=. Consider the operator (),,()DfKQQfQdQcΩ′′′≡∫∫ (13) and corresponding to it quadratic form (),(,,)()()DffKQQfQdQfQdQcΩΩ′′′==∫∫∫∫ ( )211221212exp(,)NMnNmMicnscmsfssdsds=−=− Ω=+=∑∑∫∫ 212220ccππ=≥I. Obviously that this inequality modifies into equality only as 0I=. From here follows that the kernel (),,KQQc′ is positively defined . Accordingly op-erator D is positive on nonnegative functions cone K of the space ()CΩ . According to it D leaves invariant the cone K, i.e. D⊂KK. Complex decomplexified space ()CΩ  we con-sider as a direct sum of two real spaces of continuous functions ()()()CCCΩ=Ω⊕Ω in the domain Ω. The elements of this space are written as (,)()Tfuv C=∈Ω, ()uC∈Ω, ()vC∈Ω. Norms in these spaces have the form: ()max()CQuuQΩ∈Ω=, ()max()CQvvQΩ∈Ω=, P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 68 ()()()()max,CCfuvCΩΩΩ=. The Equation (10) in the decomplexified space ()CΩ we reduce to equivalent to it system of the nonlinear eq-uations ( )122()()(,),,() ()()uQuQBuvKQQFQdQuQvQcΩ′′′′=≡ ′′+∫∫ , ( )222()()(,),,() ()()vQvQBuvKQQFQdQuQvQcΩ′′′′=≡ ′′+∫∫ . (14) Denote the closed convex set of continuous functions as ()RSC⊂Ω supposing that uvRRRSSS=⊕, {}():uRCSuuRΩ=≤, {}():vRCSvvRΩ=≤, max(,,)()QRKQQFQdQc∈Ω Ω′′′=∫∫ . Theorem 1. The operator 12(,)TBBB= determined by the Formula (14) maps a closed convex set RS of the Banach space ()CΩ in itself and it is completely con-tinuous. Proof. At first we show that :()()BCCΩ→Ω. Let (,)Tfuv= be any function belonging to ()CΩ. At ()12,ccc∈Λ the kernel (),,KQQc′ is a continuous function with respect to its arguments in the closed do-main Ω×Ω. Then accordingly to the Cantor theorem  (),,KQQc′ is a uniformly continuous function in Ω×Ω. From here follows: for any points 11(,)QQ′, 22(,)QQ′ such that whenever ()()1122,,QQQQ′′−<δ, then ()()1122,,,,KQQKQQaε′′−. Indicate that the operator ():()()TcCCΩ→Ω is completely continuous. Proof of this property is similar to the proof of a complete conti-nuity of the operator 11(,)TBBB= in the Theorem 1. According to  such values of parameters (0)(0)212(,)cc∈¡ at which linear homogeneous Equation (25) has distinct from identical zero solutions are points of possible branching of solutions of a system of nonli-near Equations (23) and (24). The eigenfunctions of (25) are used at construction branching-off solutions of (23) and (24). The spectral parameters1cand2care included non- li-nearly into the kernel of the integral operator. Therefore a problem on finding the distinct from 012(,,)fQcc solutions of (25) is a nonlinear two-parametric spectral problem. It consists in finding such values of real para-meters ()12,ccc∈Λ at which (25) has distinct from identical zero solutions. In operational form a nonlinear two-parametric prob-lem is presented as ()1212(,)(,)0ccxETccx≡−=A. (26) Here E is an identical operator and 12(,)Tcc is a linear integrated operator acting in the Banach space ()CΩ. It is necessary to find eigenvalue ()(0)(0)12,cccc=∈Λ and corresponding eigenvectors ()(0)xC∈Ω ((0)0x≠) such that (0)(0)(0)12(,)0ccx=A. By direct check we ascertain that for any values of parameters ()12,ccc ∈Λ the function ()0ˆ(,)(),,QFQKQQdQΩ′′′ϕ=∫∫cc (27) is one of eigenfunctions. Write a conjugate to (25) equation required in later ( )0()()(),,()(,)FQQTKQQQdQfQ∗Ω′′′ψ=ψ≡ψ∫∫ccc. (28) At arbitrary ()12,ccc ∈Λ the function 0ˆ()()QFQψ= . (29) is one of eigenfunctions of (28) The existence of distinct from identical zero solutions of (25) at arbitrary ()12,ccc∈Λ testifies to the exis-tence of coherent components of a spectrum contermin-ous with the domain cΛ. For finding the distinct from 0ˆ(,)Qϕc solutions we exclude from the kernel of integral Equation (25) the eigen function (27), namely: consider the equation ()(,)(),,()QTQQQdQcccΩ′′′ϕ=ϕ≡ϕ∫∫%K, (30) where () () ()( )000(),,,,(),,FQQQKQQQQfQcccc′′′′=−ψϕ′K, (31) 2000ˆ()() ˆLQQψψ=ψ, ( )()2000ˆ,,ˆLQQ′ϕ′ϕ=ϕcc. (32) P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 71From Schmidt Lemma  follows that 0(,)Qϕc will not be an eigenfunction of this equation for any val-ues ()12,ccc ∈Λ. Thus from a spectrum of operator there is excluded coherent component coinciding with the domain cΛ and corresponding to the function 0(,)Qϕc. Using the property of degeneration of the kernel ()12,,,QQcc′K, we reduce (25) to equivalent system of linear algebraic equations having coefficients analytical-ly dependent on parameters 1c, 2c. We write (25) as ( )121122(,)expNMnmnNmMssxicnscms=−=−ϕ=+−∑∑ 0012(,)xss−ψ , (33) where nmx, 0x are constants determined by the formu-las ( )1212 112201212(,) exp22(,,,)nm ccFssxicnscmsfssccΩ′′′′=−+×′′ππ∫∫ 1212(,)ssdsds′′′′×ϕ(),nNNmMM=−÷=−÷ , 0012121212(,,,)(,)xssccssdsdsΩ′′′′′′=ϕϕ∫∫ . From the Formula (33) follows, that the function 12(,)ssϕ will become known, if will be found nmx, 0x. Multiplication of both parts of (33) by ( )12 112201212(,) exp(,,,)Fssicksclsfsscc′′′′−+′′ at kNN=−÷, lMM=−÷ and by 012(,)ss′′ϕ, and integration over Ω gives a homogeneous system of the linear algebraic equations for finding nmx, 0x ( )() 12,NM klklnmnmnNmMxaccx=−=−=∑∑ ,kNNlMM=−÷=−÷. (34) Here ( )()()()() ( )12()()121212012,,,,1,klklklnmnmnmbccacctccdccdcc=−+, ( )( )121212 201212(,),(,,,)4klnm ccFsstcc fssccΩ=×π∫∫ ( )( ){}112212expicnkscmlsdsds×−+−, ( )() 121212012201212(,),(,)(,,,)4klccFssbccssfssccΩ=ψ×π∫∫ , ()112212expicksclsdsds×−+, ()()12012112212,(,)expnmdccssicnscmsdsdsΩ=ϕ+∫∫ , ()0120120121212,(,)(,,,)dccssssccdsdsΩ=ψϕ∫∫ . For coefficients of the matrix ( )1111,,MnmccaccA= ()()1111,,,,klMnmknNNmlMMccacc=−÷=−÷= the equality ()() 11,lkmnacc= ( )()11,klnmacc= is valid, i.e. MA is the Hermitian or self-adjoint matrix. Write the equivalent to (26) nonlinear two-paramet— rical spectral problem, corresponding to a system of Eq-uation (34), as ()1212(,)(,)0MMMccccxEAx≡−=A, (35) where ME is a unit matrix of dimension 22NM×. In order that the system (34) should have distinct from zero solutions, it is necessary ()1212(,)det(,)0MMccccEAΨ=−=. (36) It is easy to be convinced, that 12(,)ccΨis a real function. Really as 12(,)MccT is the Hermitian matrix then it is obvious that ()12(,)MccEA− is also the Hermitian matrix. It is known  that the determinant of the Hermitian matrix is a real number. So, 12(,)ccΨ is a real function with respect to the real arguments 1c and 2c. Therefore, the problem on finding the set of eigenva-lues of (25) or equivalent linear algebraic system (34) is reduced to finding the nulls of the function 12(,)ccΨ. Consider a necessary later on auxiliary one-dimen- sional spectral problem (as a special case of the problem (35)) on the ray 21cc=γ (γ is a real coefficient, ()12,ccc∈Λ). Introduce into consideration the matrix - function ()111(),MMccc≡γ%AA and connected with it the one-dimensional spectral problem ()()1111,(,)0MMMccccγ=−γ=xEAx%A. (37) It is easy to be convinced, that from the properties of coefficients of matrix 12(,)MccA follows, that the ma-trix function 12(,)MccA is continuous and differentia-ble on the variables in any open and limited domain 2cΛ⊂Λ⊂¡. In other words 12(,)MccA is a holo-morphic matrix - function, if 12,cc to continue into the domain of complex variables. Corresponding to (37), Equation (36) has the form ()1111(,)det(,)0MMccccΨγ=−γ=EA . (38) We denote the spectrums of the problems (35) and (37) as ()sA and ()s%A, respectively, and the parameter domain 1c as {}111:0cccaΛ=<≤. Then for proper- P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 72 ties of the spectrum of (35) the Theorem 1 from  is applied which relatively to (35) is formulated thus: Theorem 2. Let at each 12(,)cccc=∈Λ the matrix 222212(,)(,)NMNMMcc ××∈££AL be the Fredholm oper-ator with a zero index, the matrix - function ()2222(,):,NMNMc××⋅⋅Λ→ ££AL be holomorphic in the domain cΛ and 1()cs≠Λ%A. Moreover, let function ()12,ccΨ be continuously differentiable in cΛ. Then: 1) Each point of a spectrum (0)1()cs∈%A is isolated and it is eigenvalue of the matrix - function ()111(),ccc≡γ%AA , to it is corresponding the fi-nite-dimensional eigensubspace()()(0)1Nc%A and fi-nite-dimensional root subspace; 2) Each point ()(0)(0)(0)11,ccc=γ∈Λc is a point of spectrum of the matrix - function 12(,)λλA; 3) If ()2(0)(0)12,0ccc′Ψ≠ then in some vicinity of the point (0)1c there is a unique continuous differentiable function ()221ccc= solving the Equation (36), i.e. in some bicircular domain ( ){(0)(0)012111222,:,ccccccΛ=−<ε−<ε }(0)(0)012111222,:,ccccccΛ=−<ε−<ε there exists a connected component of spectrum of the matrix-function 12(,)ccA( where 1ε, 2ε are small real constants). Proof of this theorem concerning the nonlinear two-parametrical spectral problem of the type (35) for more general case (when the operators E and 1(,Tc 2)c act in the infinite dimensional Banach space) is presented in . For satisfaction of conditions of Theo-rem 1 from  it is necessary to show that the matrix - function ()12,ccA is the Fredholm matrix at 12(,)ccc∈Λ. This property follows from the known equality  ()()dimkerdimker∗=AA. The existence of connected components of spectrum of the matrix - function ()12,ccA, under condition of ()2(0)(0)12,0ccc′Ψ≠, follows from the existence theorem of implicitly given function [20, 21]. Let ()1ic be a root of (38). Then ()()()()121,iiicccc=γ∈Λ is eigenvalue of the problem (33). Consider the equation 12(,)0ccΨ= as a problem on finding the implicitly given function 221()ccc= in the vicinity of a point ()1ic for which the conditions of existence theorem  are satisfied. Hence we have the Cauchy problem 12122112(,)(,)ccccdcdccc′Ψ=− ′Ψ, (39) ()()()()211iiiccc=γ. (40) Solving numerically (39) and (40) in some vicinity of a point ()1ic, we find the i-th connected component of spectrum (spectral line) of the matrix - function 12(,)MccA. By found solutions of the Cauchy problem at the fixed values ()()()12,iicc the eigenfunctions of (25) are deter-mined through the eigenvectors of the matrix ()()()12,iiMccA obtained by the known methods. Thus four-dimensional matrix MA is reduced to two-dimensional one by means of corresponding renum-bering of elements. 4. Numerical Algorithm of Finding the Solutions of a Nonlinear Equation Show one of iterative processes for numerical finding the solutions of the system (14) based on the successive ap-proximations method : 11()(,)(,,)()nnnuQBuvKQQFQ+Ω′′=≡×∫∫c 22()()()nnnuQdQuQvQ′′×′′+, 12()(,)(,,)()nnnvQBuvKQQFQ+Ω′′=≡×∫∫c 22()()()nnnvQdQuQvQ′′×′′+ (0,1,...n=). (41) After substituting the function arg()arctg()()nnnfQvQuQ=() arg()arctg()()nnnfQvQuQ()/ (obtained on the basis of successive approxima-tions (41)) into (12), we denote the obtained sequence of function values as nI{}. For the sequence nI{} the Theorem 4.2.1 from  is fulfilled. From here follows, that the sequence nI{} is a relaxation one for the func-tional (7) and numerical sequence ()nσI{} is conver-gent. At realization of the iterative process (41) in the case of even on both arguments function 12(,)Fss and symmetric domains G and Ω it is expedient to use the property of invariance of integral operators 1(,)Buv, 2(,)Buv in the system (14) concerning the type of parity of functions 12(,)uss, 12(,)vss. The functions u, v having certain type of evenness on corresponding argu-ments belong to the appropriate invariant sets ijU, kVl of the space ()ΩC. Here the indices ,,,ijkl have P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 73values 0 or 1. In particular, if 1201(,)ussU∈ then 1212(,)(,)ussuss−= and 1212(,)(,)ussuss−=−. By direct check we are convinced that such inclusions take place: 12, ijkijijkkBUVUBUVV⊂⊂UU()()lll, ijkijkUVUV⊂BUU()ll. The possibility of existence of fixed points of the op-erator B belonging to appropriate invariant set (i.e. solutions of system (14) and, respectively, Equation (10)) follows from these relations. 5. Numerical Example Consider an example of approximation of the function ()()1212(,)cos2sinFssss=ππ (Figure 1), given in the domain (){}1212,:1,1Gssss=≤≤⊂Ω, for 221111NM×=× and values of parameters 11.6c= and 21.2c= belonging to the ray210.75cc=. The possible branching lines of solutions of the system (14) and accordingly the Equation (10), as solutions of two-dimensional spectral problem (25), are shown in Figure 2. Here the first branching lines are denoted by numbers 1 and 2. To the solutions branching-off at the points of these lines there correspond the odd on 2s functions 12arg(,)fss and the coefficients of transfor-mation ,nmI (,nNNmMM=−÷=−÷) are real, but nonsymmetrical concerning to the plane XOZ. In Figure 3 in logarithmic scale are presented values of the functional σ obtained on the solutions of two types at values of parameter 210.75cc=: the curve 1 Figure 1. The function ()( )1212(,)cos2sinFssss=ππ given in the domain ( ){}1212,:1,1Gssss=≤≤⊂Ω. Figure 2. The branching lines of solutions corresponds to solutions in a class of real functions 0()fQ, curve 2 – to the branching-off solution with odd on 2s argument ()12arg,fss. From analysis of Figure 3 follows that at the point 10.77c≈ from real solution branch-off more effective complex-conjugate between themselves solutions, on which the functional σ ac-cepts smaller values, than on the real solution. If to in-troduce into consideration parameter 22CMc= charac-terizing the quantity of basic functions in transformation (1), the identical efficiency of approximation (identical values of the functional σ on real and branching-off solutions) is reached with use of the branching-off solu-tion at decrease of the quantity of basic functions on the value 210.75Cc∆=∆. An amplitude (а) and argument (b) of approximate function are given in Figure 4 for 11.6c= and 21.2c=. The amplitude values of the Fourier Transform coefficients corresponding to this solution are shown in Figure 5. As we see in figure, the values of amplitudes of coefficients are nonsymmetrical concerning the plane Figure 3. The values of functional on initial and branch-ing-off solutions. P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 74 (a) (b) Figure 4. The modulus (а) and argument (b) of approxima-tion function. Figure 5. The optimum amplitude of Fouier transform coefficients. YOZ, but the amplitude of approximate function (Figure 4, а) is symmetric. For comparison of approximate functions, corres-ponding to different solutions of (10), the curves corres-ponding to different types of the presented solutions in the section 10s≡ are given in Figure 6. The curve 1 corresponds to the given function 2(0,)Fs, the curve 2 – to branching-off solution, the curve 3 − to real solution 02(0,)fs. Obviously that the branching-off solution bet-ter (in meaning of the functional σ) approaches the prescribed function by the module. 6. Conclusion Mark the basic features and problems arising at investi-gation of the considered class of tasks: The basic difficulty to solve this class of problems is study of nonuniqueness and branching of existing solu-tions dependent on the parameters 12,cc entering into the discrete Fourier Transform. As follows from investigations, presented, in particu-lar, in [3,17] (for a special case, when 12(,)Fss= 1122()()FsFs⋅), the quantity of the existing solutions grows considerably with increase of the parameters 12,cc. Let us indicate, that in many practical applica-tions, in particular, in the synthesis problems of radiating systems, it is important to obtain the best approximation to the given function 12(,)Fss at rather small values of parameters 12,cc. This allows limiting by investigation of several first points (lines) of branching. To find the branching points (lines) of solutions of (8), it is necessary, as opposed to [3, 17], to solve not enough studied multiparametric spectral problem. The offered in this work approaches allow to find the solutions of a nonlinear two-parametric spectral problem for homoge-neous integral equations with degenerate kernels analyt-ically dependent on two spectral parameters. 00.20.40.60.811.2-1 -0.5 00.5 1123| f |s2 Figure 6. The given (curve 1) and approximation functions in the section 20s= corresponding to branching-off (curve 2) and real (curve 3) solutions. P. SAVENKO ET AL. Copyright © 2010 SciRes. AM 75When finding the solutions to a system of Equation (14) by successive approximations method, to obtain the solutions of a certain type of parity of the function ()12arg,fssit is necessary to choose an initial approxi-mation ()012arg,fss of the same type of parity ac-cording to (42). To obtain the irrefragable answer concerning the branching-off solutions for certain values of parameters 12,cc it is necessary to use the branching theory of so-lutions . It is the object of special investigations. 7. References  B. M. Minkovich and V. P. Jakovlev, “Theory of Synthesis of Antennas, ” Soviet Radio, Moscow, 1969.  P. A. Savenko, “Numerical Solution of a Class of Nonlinear Problems in Synthesis of Radiating Systems,” Computational Mathematics and Mathematical Physics, Vol. 40, No. 6, 2000, pp. 889-899.  P. 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