Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform

The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear problem 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.


Introduction
The mean-square approximation of real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on physical parameters, is widely used, in particular, at modeling and solution of the synthesis problems of different types of antenna arrays, signal processing etc. [1][2][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 explored nonlinear spectral two-parametric problem.The methods of investigation and numerical finding the solutions of one-parametric spectral problems at presence of discrete spectrum [4][5][6][7][8] are most well-developed.The existence of coherent components of spectrum, which are spectral lines for the case of real parameters [9], is essential difference of nonlinear two-parametric spectral problems.
In the work a variational problem on the best meansquare approximation of a real finite nonnegative function 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 spectrum of holomorphic matrix functions dependent on two spectral parameters is proved.It justifies the application of implicit functions methods to multiparametric spectral problems [9].The applicability of this theorem to the analysis of spectrum of two-dimensional integral homogeneous equation to which is reduced the problem on finding the lines of possible branching of solutions of the Hammerstein equation, is shown.Algorithms for numerical finding the optimum solutions of an approximation problem are constructed and justified.Numerical examples are presented.

Problem Formulation, Basic Equations and Relations
Consider the special case of double discrete Fourier transform ( ) , 1211221212 ,,, C fffssfssdsds , C fff Denote an augmented space of continuous functions with entered scalar product and norm (3) as (2)   () C Ω and notice that its augmentation coincides with the Hilbert space 2 () L Ω [10].By direct check we are sure that such equality ( ) is valid.From here follows, that A is isometric operator in sense (4).Using the entered scalar products (2) and (3) we find the conjugate operator required later on (,),(,), (,) 0,(,)\, FssssG Fss ssG On the basis of necessary condition of functional minimum we obtain a nonlinear system of equations relating to the components of vector I in the space I H that are represented in the vector and expanded forms, respectively: ( ) ,exparg 4 Acting on both parts of ( 9) by operator A we obtain equivalent to (9)   The equivalence of ( 9) and (10) follows from the following lemma.
Lemma 1. Between solutions of Equations ( 9) and (10) there exists bijection, i.e., if I * is a solution of (9) then fA I * * = is the solution of (10); on the contrary, if f * is the solution of (10) then is the solution of (9).
Proof.Let I * be a solution of (9).Then solves the Equation (9).Lemma is proved.
Thus owing to the equivalence of ( 9) and ( 10) we consider 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 operator A is a set of continuous functions in the domain Ω belonging to the space ( ) 2 L Ω and this set is a compact in the space 2 () L Ω [12], we shall investigate solutions of (10) in the space () C Ω .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).
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 Obviously that this inequality modifies into equality only as 0 I = .From here follows that the kernel ( ) ,, KQQ c ′ is positively defined [13].Accordingly operator D is positive on nonnegative functions cone K of the space () C Ω [14].According to it D leaves invariant the cone K , i.e.D ⊂ KK .Complex decomplexified space () C Ω [10] we consider as a direct sum of two real spaces of continuous functions ()()() CC C Ω=Ω⊕Ω in the domain Ω .The elements of this space are written as (,)() T fuv C =∈Ω , () uC ∈Ω , () vC ∈Ω .Norms in these spaces have the form: The Equation (10) in the decomplexified space () C Ω we reduce to equivalent to it system of the nonlinear equations ( ) () ()(,),,() ()() Denote the closed convex set of continuous functions as () .On this basis we obtain () 0 ()()(),() Thus, from ( 16) and ( 17) follows 111122 () 0, 0 lim(,)(,) We show that a set of functions gR SS B = satisfies conditions of the Arzela theorem [12] Theorem is proved.
From the Theorem 1 follows satisfaction of conditions of the Schauder principle [16] according to which the operator 12 (,) T BB B = has a fixed point (,) T fuv * * * = belonging to the set R S .This point is a solution of a system of Equation ( 14) and Equation (10), respectively.Substituting (,) T fuv * * * = into (12), we obtain a solution of ( 9) being a stationary point of the functional (7).
The solutions of a system of equations analogous with (14) in a case of one-dimensional domains Ω were investigated for the synthesis problem of linear antenna array in particular in [17].The obtained there results show that for equations of the type ( 10) and ( 14) non-uniqueness and branching of solutions dependent on the size of physical parameter are characteristic.Directly the results [17] cannot be transferred on the two-dimensional two-parametric problem ( 8) and ( 14).Here, as unlike the points of branching [17], the branching lines of solutions exist and a problem on finding the lines of branching is a nonlinear two-parametrical spectral problem.
Easily to be convinced that function is one of solutions of (10) in the class of real functions.Since, as shown before, the operator D determined by (13), is positive on the nonnegative functions cone C() ∈Ω K , D ⊂ KK and F ⊂ K , then 0 fDF = also is a nonnegative function in the domain Ω .
To find the lines of branching and complex solutions of ( 10 Notice the properties of integrand in the system (14).They are continuous functions with respect to the arguments.After substitution (20) and ( 21) into ( 14) the integrand develop in equiconvergent power series by functional arguments w and ω , numerical parameters µ and ν in the vicinity of a point ( ) ( ) ,, mnpq BQQ c ′ are coefficients of expansion continuously dependent on the arguments.Substituting ( 20) and ( 22) into ( 14) and taking into account that , fQ c ′ solves the system ( 14) we obtain a system of nonlinear equations with respect to small solutions w , ω :

Nonlinear Two-Parametric Spectral Problem
For further application of methods of the branching theory of solutions of nonlinear equations [18] to a system ( 23) and (24) it is necessary to find solutions of distinct from trivial of the linear homogeneous integral equation obtained equating to zero the left part of (24) ( ) ),,,()  (,,) in the Theorem 1.According to [18] such values of parameters (0)(0)2 12 (,) cc∈ ¡ at which linear homogeneous Equation (25) has distinct from identical zero solutions are points of possible branching of solutions of a system of nonlinear Equations ( 23) and (24).The eigenfunctions of (25) are used at construction branching-off solutions of ( 23) and (24).
The spectral parameters 1 c and 2 c are included non-linearly 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 parameters ( )

,
c cc ∈Λ at which (25) has distinct from identical zero solutions.
In operational form a nonlinear two-parametric problem is presented as Here E is an identical operator and  ,,,,(), , From Schmidt Lemma [18] follows that 0 (,) Q ϕ c will not be an eigenfunction of this equation for any values ( )    YOZ , but the amplitude of approximate function (Figure 4, а) is symmetric.
For comparison of approximate functions, corresponding to different solutions of (10), the curves corresponding to different types of the presented solutions in the section 1 0 s ≡ 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 better (in meaning of the functional σ ) approaches the prescribed function by the module.

Conclusion
Mark the basic features and problems arising at investigation of the considered class of tasks: The basic difficulty to solve this class of problems is study of nonuniqueness and branching of existing solutions dependent on the parameters 12 , cc entering into the discrete Fourier Transform.
As follows from investigations, presented, in particular, in [3,17] (for a special case, when 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 homogeneous integral equations with degenerate kernels analytically dependent on two spectral parameters.
cc =∆ % , 22 y cc =∆ %.If it is necessary for the accepted assumptions we shall consider the formula Fss is a real continuous and nonnegative in the domain G function.Consider a problem on the best mean-square approximation of the function 12 (,) Fss in the domain Ω by the module of double discrete Fourier transform (1) owing to select of coefficients of the vector I .We shall formulate it as a minimization problem of the functional into account (4) and (5), we write the functional () I σ in a simplified form

2 L
set of its nulls consists of only null element from the last identity follows, what f AfH I * * =∈ % is a solution of (10).On the contrary, let f fH * ∈ % solves the Equation (10).The operator A * acts from the space 2 * coincides with the space 2 L [10].From here follows, that A * acts from the space 2 account that F is a finite function determined by (6), and f * is continuous, the function ( ) exparg() Fif * is quadratic integrability in the domain Ω , i.e. is the result of action of operator A on an element I * , i.e. and taking into account that a set of operator nulls consists of only a null element we obtain 12 arg(,) fss on two arguments (or on that argument on which 12 (,) Fss is an even function).Below taking into account the property 2) for uniqueness of solutions we set the parameter 0 β= .
(14) maps a closed convex set R S of the Banach space ()C Ω in itself and it is completely continuous.Proof.At first we show that :()() BCC Ω→Ω .Let (,) T fuv = be any function belonging to () function with respect to its arguments in the closed domain Ω×Ω.Then accordingly to the Cantor theorem[15] continuous function in Ω×Ω .From here follows: for any points 11 (,) QQ ′ , 22 (,) QQ ′ such that whenever ( ), branching-off from real solution 0 (,) fQ c , we consider a problem on finding such set of values of paωµν on parameters µ and ν .
fQc ′ > .Indicate that the operator ():()() T cCC Ω→Ω is completely continuous.Proof of this property is similar to the proof of a complete continuity of the operator 11 (,) T BB B = Tcc is a linear integrated operator acting in the Banach space () C Ω .It is necessary to find eigenvalue() eigenfunctions of (28) The existence of distinct from identical zero solutions of (25) at arbitrary ( )12 ,c cc ∈ Λ testifies to the existence of coherent components of a spectrum conterminous with the domain c Λ .For finding the distinct from 0 ˆ(,) Q ϕ c solutions we exclude from the kernel of integral Equation (25) the eigen function (27), namely: consider the equation

Figure 2 . 5 .Figure 3 .
Figure 2. The branching lines of solutions corresponds to solutions in a class of real functions 0 () fQ, curve 2to the branching-off solution with odd on 2 s argument
quantity of the existing solutions grows considerably with increase of the parameters 12 , cc .Let us indicate, that in many practical applications, 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 ,