Numerical Approximation of Real Finite Nonnegative Function by the

Applied Mathematics, 2010, 1, 65-75 doi:10.4236/am.2010.11008 Published Online May 2010 (http://www.SciRP.org/journal/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 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. 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 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. [1-3]. 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 [4-8] 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 Copyright © 2010 SciRes.

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.

2. Problem Formulation, Basic Equations and Relations Consider the special case of double discrete Fourier transform f ( s1 , s2 ) =

N2

M 2 ( n)

∑

∑

n =− N1 m =− M 1 ( n )

I nm exp i ( c%1 xnm s1 + c%2 ynm s2 )

setting here xnm = n∆ x ( n = − N ÷ N ) , ynm = m∆ y (m = − M ÷ M ) ; c1 = c%1∆ x , c2 = c%2 ∆ y . If it is necessary for the accepted assumptions we shall consider the formula f ( s1 , s2 ) = AI ≡

N

M

∑ ∑

n =− N m =− M

I nm exp i ( c1ns1 + c2 ms2 )  (1)

as a linear operator, acting from complex finite-dimensional N ×M space H I = £ 2 2 ( N 2 = 2 N + 1 , M 2 = 2 M + 1 ) into the space of complex-valued continuous functions with respect to two real variables determined in the domain Ω = {( s1 , s2 ) : s1 ≤ π c1 , s2 ≤ π c2 } .

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Here c1 , c2 are any real non-dimensional numerical parameters belonging to Λc = {( c1 , c2 ) : 0 < c1 ≤ a, 0 < c2 ≤ b} . The function f ( s1 , s2 ) is 2π c1 - periodic function on argument s1 and 2π c2 - periodic on s2 . In considered spaces we introduce scalar products and generable by them norms 4π2 N M ( I1 , I 2 )H I = ∑ ∑ I nm I nm , c1c2 n =− N m =− M I = ( I, I ) H

1/ 2

( f1 , f 2 )C(

(2) Ω)

,

(2)

I

= ∫∫ f1 ( s1 , s2 ) f 2 ( s1 , s2 )ds1ds2 , Ω

f = ( f , f )C (2) . 1/ 2

(3)

(Ω )

Denote an augmented space of continuous functions with entered scalar product and norm (3) as C(2) ( Ω ) and notice that its augmentation coincides with the Hilbert space L2 (Ω) [10]. By direct check we are sure that such equality AI

2

= ∫∫ f ( s1 , s2 ) ds1ds2 = ∑ I nm 2

Ω

2

= I

2

(4)

n,m

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 A∗ f =

c1c2 4π2

∫∫ f ( s1 , s2 ) exp  −i ( c1ns1 + c2 ms2 ) ds1ds2 Ω

( n = − N ÷ N , m = −M ÷ M ) .

(5)

Let such function be given  F ( s , s ), F% ( s1 , s2 ) =  1 2 0,

( s1 , s2 ) ∈ G ⊆ Ω, ( s1 , s2 ) ∈Ω \ G ,

(6)

where F (s1 , s2 ) is a real continuous and nonnegative in the domain G function. Consider a problem on the best mean-square approximation of the function F (s1 , s2 ) 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 σ( I ) = F − A I

2

≡ F− f

2

(7)

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 H I that are represented in the vector and expanded forms, respectively:

{

}

I = A∗ F exp i arg ( AI ) , I nm =

c1c2 4π2

 



N

(9) M

∑ I nm × ∫∫ F ( s1 , s2 ) exp i arg  k∑ =− N l =− M Ω

)

 

}

× exp i ( c1ks1 + c2 ls2 ) − ( c1ns1 + c2 ms2 ) ds1ds2 

( n = − N ÷ N , m = −M ÷ M ) .

(9′)

Acting on both parts of (9) by operator A we obtain equivalent to (9) the Hammerstein type nonlinear integral equation relating to f : f (Q) = Bf ≡ ∫∫ K ( Q, Q′, c ) F (Q′)exp [ i arg f (Q′)] dQ′ , (10) Ω

where Q′ = ( s1′ , s2′ ) , dQ′ = ds1′ds2′ , c = ( c1 , c2 ) ; K ( Q, Q ′, c ) = K1 ( s1 , s1′ , c1 ) ⋅ K 2 ( s2 , s2′ , c2 ) , K1 ( s1 , s1′ , c1 ) =

(11)

c1 N ∑ exp ic1n ( s1 − s1′ ) ≡ 2π n =− N

N c  sin  2 1 ( s1 − s1′ ) c  2 , ≡ 1 c1 2π ( s1 − s1′ ) 2 K 2 ( s2 , s2′ , c2 ) =

c2 M ∑ exp ic2 m ( s2 − s2′ ) ≡ 2π m =− M

M c  sin  2 2 ( s2 − s2′ ) c2 2  . ≡ c2 2π ( s2 − s2′ ) 2 Note, that the kernel (11) of Equation (10) is degenerate and real. We shall consider one of the properties of function exp ( i arg f (Q′) ) entering into (10) at f (Q′) → 0 . Obviously that the function exp (i arg f (Q ′) ) =

f (Q ′) u (Q ′) + iv(Q′) ≡ 12 f (Q ′) u 2 (Q′) + v 2 (Q′)

(

)

is continuous if u (Q′) = Re f (Q′) and v (Q ′) = Im f (Q′) v (Q′) = Im f (Q′) are continuous functions, where exp ( i arg f (Q ) 1 in the Hilbertian space H I . Taking into account (4) and (5), we write the functional σ(I ) in a simplified form exp i arg f (Q′) ) = 1 for any f (Q′) . If u (Q′) → 0 and v (Q′) → 0 C(2) ( Ω)

σ (I ) = F

2 (2) C ( Ω)

− 2 ( F , AI )

Copyright © 2010 SciRes.

C(2) ( Ω)

2 (2) C ( Ω)

+ I

2 HI

.

(8)

simultaneously then f (Q′) → 0 is a complex zero. Its argument is undetermined accordingly to definition of AM

P. SAVENKO ET AL.

complex zero [11, p. 20]. On this basis we redefine exp ( i arg f (Q′) ) at u (Q′) → 0 and v (Q′) → 0 as a function which has module equal to unit and undetermined argument. 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 f ∗ = AI ∗ is the solution of (10); on the contrary, if f∗ is the solution of (10) then I∗ = A∗ {F exp [i arg( f∗ )]}

(12)

67

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 L2 ( Ω ) and this set is a compact in the space L2 (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 f (Q) is a solution of (10) then the conjugate complex function f (Q) is also the solution of (10). 2) If function f (Q) is a solution of (10), then

exp ( iβ ) f (Q) is also the solution of (10) ( β is any real i AI∗ ≡ Proof. Let I∗ be a solution of (9). Then I∗ − A∗ F constant). 3) For even on two arguments (or on one argument) I∗ − A∗ F exp  i arg ( AI∗ )  ≡ 0 . Acting on this identity by the functions F (s1 , s2 ) the nonlinear operator B that is in ∗ part of (10), is an invariant concerning the type 0 linear operator A , we have AI ∗ − AA F exp  i arg ( AI∗ the≡ right of parity of the function arg f (s1 , s2 ) on two arguments F exp i arg ( AI∗ ) ≡ 0 . Since the operator A acts from the space (or on that argument on which F (s1 , s2 ) is an even N2 ×M 2 (2) function). HI = £ into the space C( Ω ) and accordingly into Below taking into account the property 2) for uniquethe space H% f = L2 (Ω) , and the set of its nulls consists ness of solutions we set the parameter β = 0 . of only null element from the last identity follows, what Consider the operator AI∗ = f ∗ ∈ H% f is a solution of (10). Df ≡ ∫∫ K ( Q, Q′, c ) f (Q′)dQ′ (13) On the contrary, let f∗ ∈ H% f solves the Equation (10). Ω is the solution of (9).

{

}

{

}

The operator A∗ acts from the space H% ∗f = L∗2 (Ω) into the space H I = £ space

N2 ×M 2

and corresponding to it quadratic form

( Df , f ) = ∫∫ ∫∫ K (Q, Q′, c) f (Q′)dQ ′ f (Q)dQ =

[10] and the Hilbertian

L∗2

coincides with the space L2 [10]. From here follows, that A∗ acts from the space H% = L (Ω) into f

2

the space H I = £ N2 ×M 2 . Taking into account that F is a finite function determined by (6), and f∗ is conti-

Ω Ω

=

N

M

∑ ∑

2

∫∫ exp i ( c1ns1 + c2 ms2 )  f (s1 , s2 ) ds1ds2  2π  2π  =    I  c1  c2 

nuous, the function F exp ( i arg( f ∗ ) ) is quadratic integrability in the domain Ω , i.e. F exp ( i arg( f∗ ) ) ∈ H f .

Thus A∗ ( F exp ( i arg( f ∗ ) ) ) = I∗ ∈ H I and the right part

of (10) is the result of action of operator A on an element I∗ , i.e. AI ∗ = AA∗ ( F exp ( i arg( f∗ ) ) ) = f∗ . Writ-

(

)

ing this equality as A I∗ − A∗ ( F exp ( i arg( AI∗ ) ) ) = 0 and taking into account that a set of operator nulls consists of only a null element we obtain I∗ = A∗ ( F exp ( i arg( AI ∗ ) ) ) . So, I∗ = A∗ ( F exp ( i arg( f∗ )

A F exp i arg( f∗ ) ) ) 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. Copyright © 2010 SciRes.

=

n =− N m =− M Ω

2

≥ 0.

Obviously that this inequality modifies into equality only as I = 0 . From here follows that the kernel K ( Q, Q ′, 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. DK ⊂ K . Complex decomplexified space C(Ω) [10] we consider as a direct sum of two real spaces of continuous functions C(Ω) = C (Ω) ⊕ C (Ω) in the domain Ω . The elements of this space are written as f = (u, v )T ∈ C(Ω) , u ∈ C (Ω) , v ∈ C (Ω) . Norms in these spaces have the form: u

C ( Ω)

= max u (Q) , v Q∈Ω

C (Ω )

= max v (Q) , Q∈Ω

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P. SAVENKO ET AL.

f

C (Ω)

(

= max u

C ( Ω)

, v

C ( Ω)

).

The Equation (10) in the decomplexified space C(Ω) we reduce to equivalent to it system of the nonlinear equations u(Q′) u(Q) = B1 (u, v) ≡ ∫∫ K (Q, Q′, c ) F (Q′) dQ′ , u2 (Q′) + v2 (Q′) Ω v(Q) = B2 (u, v) ≡ ∫∫ K ( Q, Q′, c) F (Q′) Ω

v(Q′) u (Q′) + v2 (Q′) 2

dQ′ .

{

{

S Rv = v : v

C ( Ω)

C (Ω)

}

≤R ,

to S R . It is necessary to show that Bf1 − Bf 2 f1 − f 2

as

C (Ω)

u2

}

≤R ,

C ( Ω)

f = (u , v)T be any function belonging to C(Ω) . At

where

Ω × Ω . From here follows: for any points (Q1 , Q′1 ) ,

since

( Q1 , Q1′ ) − ( Q2 , Q2′ ) < δ , ε , where a = K ( Q1 , Q1′, c ) − K ( Q2 , Q2′ , c )
0 . Indicate that the operator T (c) : C(Ω) → C(Ω) is completely continuous. Proof of this property is similar to the proof of a complete continuity of the operator B = ( B1 , B1 )T in the Theorem 1. According to [18] such values of parameters (c1(0) , c2(0) ) ∈ ¡ 2 at which linear homogeneous Equation (25) has distinct from identical zero solutions are points Copyright © 2010 SciRes.

)

c = c1(0) , c2(0) ∈ Λ c

and corresponding eigenvectors

x (0) ∈ C ( Ω ) ( x (0) ≠ 0 ) such that A (c1(0) , c2(0) ) x (0) = 0 . By direct check we ascertain that for any values of parameters ( c1 , c2 ) ∈ Λ c the function

µ p ν q ∫∫ Bmnpq Q, Q′, c(0) wm (Q′)ωn (Q′)dQ′ , (24)

where

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 c1 and c2 are included non- linearly into the kernel of the integral operator. Therefore a problem on finding the distinct from f 0 (Q, c1 , c2 ) solutions of (25) is a nonlinear two-parametric spectral problem. It consists in finding such values of real parameters ( c1 , c2 ) ∈ Λc at which (25) has distinct from identical zero solutions. In operational form a nonlinear two-parametric problem is presented as (26) A (c1 , c2 ) x ≡ ( E − T (c1 , c2 ) ) x = 0 .

ϕˆ 0 (Q, c) = ∫∫ F (Q′) K ( Q, Q′, c ) dQ′

(27)

Ω

is one of eigenfunctions. Write a conjugate to (25) equation required in later ψ(Q) = T ∗ (c)ψ ≡ At arbitrary

F (Q ) K ( Q, Q′, c ) ψ(Q′)dQ′ . (28) f 0 (Q, c) ∫∫ Ω

( c1 , c2 ) ∈ Λ c

the function

ψˆ 0 (Q) = F (Q) .

(29)

is one of eigenfunctions of (28) The existence of distinct from identical zero solutions of (25) at arbitrary ( c1 , c2 ) ∈ Λ c testifies to the existence of coherent components of a spectrum conterminous with the domain Λ c . For finding the distinct from ϕˆ 0 (Q, c) solutions we exclude from the kernel of integral Equation (25) the eigen function (27), namely: consider the equation ϕ(Q, c) = T% (c)ϕ ≡ ∫∫ K ( Q, Q′, c ) ϕ(Q′)dQ′ ,

(30)

Ω

where F (Q ′) K ( Q, Q′, c ) − ψ0 (Q)ϕ0 ( Q′, c ) , f 0 (Q ′, c ) (31) ˆ 0 (Q ) ψ ϕˆ 0 (Q′, c ) , ϕ0 ( Q′, c ) = . (32) ψ 0 (Q ) = ψˆ 0 L ϕˆ 0 L

K ( Q, Q′, c ) =

2

2

AM

P. SAVENKO ET AL.

71

From Schmidt Lemma [18] follows that ϕ0 (Q, c) d0 ( c1 , c2 ) = ∫∫ ψ0 (s1 , s2 )ϕ0 ( s1 , s2 , c1 , c2 )ds1ds2 . Ω will not be an eigenfunction of this equation for any values ( c1 , c2 ) ∈ Λ c . Thus from a spectrum of operator For coefficients of the matrix A M ( c1 , c1 ) = anm c1 , c1 there is excluded coherent component coinciding with ( kl ) ( lk ) the equality amn ( c1 , c1 ) = the domain Λ c and corresponding to the function M c1 , c1 = anm ( c1 , c1 ) k , n =− N ÷ N m , l =− M ÷ M ϕ0 (Q, c) . ( kl ) = anm ( c1 , c1 ) is valid, i.e. A M is the Hermitian or Using the property of degeneration of the kernel self-adjoint matrix. K ( Q, Q′, c1 , c2 ) , we reduce (25) to equivalent system of Write the equivalent to (26) nonlinear two-paramet— linear algebraic equations having coefficients analyticalrical spectral problem, corresponding to a system of Eqly dependent on parameters c1 , c2 . We write (25) as uation (34), as ϕ(s1 , s2 ) =

N

M

∑ ∑

n =− N m=− M

xnm exp i ( c1ns1 + c2 ms2 ) −

− x0 ψ 0 ( s1 , s2 ) ,

A (33)

where xnm , x0 are constants determined by the formulas c c F ( s1′, s2′ ) exp  −i ( c1 ns1′ + c2 ms2′ ) × xnm = 1 2 ∫∫ 2π 2π Ω f 0 ( s1′ , s2′ , c1 , c2 ) ×ϕ( s1′, s2′ )ds1′ds2′ ( n = − N ÷ N , m = − M ÷ M ) , x0 = ∫∫ ϕ0 ( s1′ , s2′ , c1 , c2 )ϕ(s1′, s2′ )ds1′ds2′ . Ω

From the Formula (33) follows, that the function ϕ(s1 , s2 ) will become known, if will be found xnm , x0 . Multiplication of both parts of (33) by F ( s1′, s2′ ) exp  −i ( c1 ks1′ + c2 ls2′ ) at k = − N ÷ N , f 0 (s1′ , s2′ , c1 , c2 ) l = − M ÷ M and by ϕ0 (s1′, s2′ ) , and integration over Ω gives a homogeneous system of the linear algebraic equations for finding xnm , x0 xkl =

 k = −N ÷ N , ( kl ) anm ( c1 , c2 ) xnm  . n =− N m =− M  l = −M ÷ M  N

M

∑ ∑

(34)

Here

kl  ( kl )  b ( ) ( c1 , c2 ) ( kl ) anm d nm ( c1 , c2 ) , ( c1 , c2 ) = tnm ( c1 , c2 ) − 1 + d 0 ( c1 , c2 )   c1c2 F ( s1 , s2 ) ( kl ) t nm ( c1 , c2 ) = 2 ∫∫ × 4π Ω f 0 ( s1 , s2 , c1 , c2 )

{

}

× exp i  c1 ( n − k ) s1 + c2 ( m − l ) s2  ds1ds2 , b( kl ) ( c1 , c2 ) =

c1c2 4π2

F ( s1 , s2 )

∫∫ f0 ( s1 , s2 , c1 , c2 ) ψ 0 (s1 , s2 ) × , Ω

× exp  −i ( c1ks1 + c2 ls2 ) ds1ds2 , dnm ( c1 , c2 ) = ∫∫ ϕ0 ( s1 , s2 ) exp i ( c1ns1 + c2 ms2 )  ds1ds2 , Ω

Copyright © 2010 SciRes.

M

(c1 , c2 )x ≡ ( EM − A M (c1 , c2 ) ) x = 0 ,

(35)

where EM is a unit matrix of dimension N 2 × M 2 . In order that the system (34) should have distinct from zero solutions, it is necessary Ψ (c1 , c2 ) = det ( E M − A M (c1 , c2 ) ) = 0 .

(36)

It is easy to be convinced, that Ψ (c1 , c2 ) is a real function. Really as TM (c1 , c2 ) is the Hermitian matrix then it is obvious that ( E − A M (c1 , c2 ) ) is also the Hermitian matrix. It is known [19] that the determinant of the Hermitian matrix is a real number. So, Ψ (c1 , c2 ) is a real function with respect to the real arguments c1 and c2 . Therefore, the problem on finding the set of eigenvalues of (25) or equivalent linear algebraic system (34) is reduced to finding the nulls of the function Ψ (c1 , c2 ) . Consider a necessary later on auxiliary one-dimensional spectral problem (as a special case of the problem (35)) on the ray c2 = γc1 ( γ is a real coefficient,

( c1 , c2 ) ∈ Λ c ). Introduce into consideration the matrix function A% M (c1 ) ≡ A M ( c1 , γc1 ) and connected with it the one-dimensional spectral problem A% M ( c1 , γc1 ) x = ( EM − A M (c1 , γc1 ) ) x = 0 .

(37)

It is easy to be convinced, that from the properties of coefficients of matrix A M (c1 , c2 ) follows, that the matrix function A M (c1 , c2 ) is continuous and differentiable on the variables in any open and limited domain Λ ⊂ Λ c ⊂ ¡ 2 . In other words A M (c1 , c2 ) is a holomorphic matrix - function, if c1 , c2 to continue into the domain of complex variables. Corresponding to (37), Equation (36) has the form Ψ (c1 , γc1 ) = det ( EM − A M (c1 , γc1 ) ) = 0 .

(38)

We denote the spectrums of the problems (35) and (37) as s (A ) and s ( A% ) , respectively, and the parameter domain c1 as Λc1 = { c1 : 0 < c1 ≤ a} . Then for properAM

: c1 − c1(0)

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P. SAVENKO ET AL.

ties of the spectrum of (35) the Theorem 1 from [9] is applied which relatively to (35) is formulated thus: Theorem 2. Let at each c = (c1 , c2 ) ∈ Λc the matrix A M (c1 , c2 ) ∈L (£ N2 ×M 2 , £ N2 × M 2 ) be the Fredholm operator with a zero index, the matrix - function

(

A (⋅, ⋅) : Λc →L £

N2 ×M 2

,£

N2 ×M 2

)

be holomorphic in the

domain Λc and s (A% ) ≠ Λ c1 . Moreover, let function Ψ ( c1 , c2 ) be continuously differentiable in Λc . Then: 1) Each point of a spectrum c1(0) ∈ s(A% ) is isolated and it is eigenvalue of the matrix - function A% (c1 ) ≡A ( c1 , γc1 ) , to it is corresponding the fi-

( ( ))

nite-dimensional eigensubspace N A% c1(0) nite-dimensional root subspace;

(

and fi-

)

2) Each point c (0) = c1(0) , γc1(0) ∈ Λc is a point of spectrum of the matrix - function A (λ1 , λ 2 ) ;

(

)

3) If Ψ ′c2 c1(0) , c2(0) ≠ 0 then in some vicinity of the point c1(0) there is a unique continuous differentiable

function c2 = c2 ( c1 ) solving the Equation (36), i.e. in

Ψ ′c (c1 , c2 ) dc2 , =− 1 dc1 Ψ ′c2 (c1 , c2 )

(39)

c2( i ) ( c1(i ) ) = γc1( i ) .

(40)

Solving numerically (39) and (40) in some vicinity of a point c1( i ) , we find the i -th connected component of spectrum (spectral line) of the matrix - function A M (c1 , c2 ) . By found solutions of the Cauchy problem at the fixed values ( c1(i ) , c2(i ) ) the eigenfunctions of (25) are determined through the eigenvectors of the matrix A M ( c1(i ) , c2(i ) ) obtained by the known methods. Thus AM is reduced to four-dimensional matrix two-dimensional one by means of corresponding renumbering 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 approximations method [2]:

{( c , c ) : c − c

′ ′ < ε1 , c2 − c2(0) < εu2 n+1 (Q) = B1 (un , vn ) ≡ ∫∫ K (Q, Q , c) F (Q ) × Ω u n (Q′) × dQ′ , < ε1 , c2 − c2(0) < ε2 there exists a connected component of un2 (Q′) + vn2 (Q′) spectrum of the matrix-function A (c1 , c2 ) ( where ε1 , vn+1 (Q) = B2 (un , vn ) ≡ ∫∫ K (Q, Q′, c) F (Q′) × ε 2 are small real constants). Ω vn (Q′) Proof of this theorem concerning the nonlinear × dQ′ ( n = 0, 1, ... ). (41) two-parametrical spectral problem of the type (35) for un2 (Q′) + vn2 (Q′) more general case (when the operators E and T (c1 , After substituting the function arg f n (Q ) = arctg (vn (Q ) u n (Q )) c2 ) act in the infinite dimensional Banach space) is arg f n (Q ) ofarctg (vn (Q )/un (Q)) (obtained on the basis of successive approximapresented in [9]. For satisfaction of conditions Theotions (41)) into (12), we denote the obtained sequence of rem 1 from [9] it is necessary to show that the matrix function values as {I n } . For the sequence {I n } the function A ( c1 , c2 ) is the Fredholm matrix at Theorem 4.2.1 from [3] is fulfilled. From here follows, (c1 , c2 ) ∈ Λc . This property follows from the known that the sequence {I n } is a relaxation one for the funcequality [19] dim ( ker A ) = dim ker A ∗ . tional (7) and numerical sequence {σ(I n )} is converThe existence of connected components of spectrum gent. of the matrix - function A ( c1 , c2 ) , under condition of At realization of the iterative process (41) in the case of even on both arguments function F ( s1 , s2 ) and Ψ ′c2 c1(0) , c2(0) ≠ 0 , follows from the existence theorem symmetric domains G and Ω it is expedient to use of implicitly given function [20, 21]. (i) (i ) (i ) (i) the property of invariance of integral operators B1 (u , v ) , c , c = γ c ∈ Λ Let c1 be a root of (38). Then 1 2 1 c B2 (u, v) in the system (14) concerning the type of parity is eigenvalue of the problem (33). Consider the equation of functions u ( s1 , s2 ) , v ( s1 , s2 ) . The functions u , v Ψ (c1 , c2 ) = 0 as a problem on finding the implicitly c = c ( c ) ( i ) having certain type of evenness on corresponding argugiven function 2 2 1 in the vicinity of a point c1 ments belong to the appropriate invariant sets U ij , Vk l for which the conditions of existence theorem [21] are of the space C( Ω) . Here the indices i , j , k , l have satisfied. Hence we have the Cauchy problem some bicircular domain Λ 0 =

}

1

(

(

2

1

(0) 1

)

)

(

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)

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values 0 or 1. In particular, if u ( s1 , s2 ) ∈ U 01 then u ( − s1 , s2 ) = u ( s1 , s2 ) and u ( s1 , − s2 ) = − u ( s1 , s2 ) . By direct check we are convinced that such inclusions take place: B1(U ij U Vk l ) ⊂ U ij , B2 (U ij U Vk l ) ⊂ Vk l , B(U ij U Vk l ) ⊂ U ij U Vk l . The possibility of existence of fixed points of the operator B belonging to appropriate invariant set (i.e. solutions of system (14) and, respectively, Equation (10)) follows from these relations.

5. Numerical Example Figure 2. The branching lines of solutions

Consider an example of approximation of the function F ( s , s ) = cos ( πs 2 ) sin ( πs ) (Figure 1), given in 1

the

2

domain

G = {( s1 , s2 ) : s1 ≤ 1, s2 ≤ 1} ⊂ Ω , for 1

2

N 2 × M 2 = 11× 11 and values of parameters c1 = 1.6 and c2 = 1.2 belonging to the ray c2 = 0.75c1 . 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 s2 functions arg f (s1 , s2 ) and the coefficients of transformation I n, m ( n = − N ÷ N , m = − M ÷ M ) 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 c2 = 0.75c1 : the curve 1

Figure 1. The function

F ( s1 , s2 ) = cos ( πs1 2) sin ( πs2 )

given in the domain G = {( s1 , s2 ) : s1 ≤ 1, s2 ≤ 1} ⊂ Ω . Copyright © 2010 SciRes.

corresponds to solutions in a class of real functions f 0 (Q ) , curve 2 – to the branching-off solution with odd on s2 argument arg f ( s1 , s2 ) . From analysis of Figure

3 follows that at the point c1 ≈ 0.77 from real solution branch-off more effective complex-conjugate between themselves solutions, on which the functional σ accepts smaller values, than on the real solution. If to introduce into consideration parameter C2 = Mc2 characterizing 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 solution at decrease of the quantity of basic functions on the value ∆C2 = 0.75∆c1 . An amplitude (а) and argument (b) of approximate function are given in Figure 4 for c1 = 1.6 and c2 = 1.2 . 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 branching-off solutions.

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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 s1 ≡ 0 are given in Figure 6. The curve 1 corresponds to the given function F (0, s2 ) , the curve 2 – to branching-off solution, the curve 3 − to real solution f 0 (0, s2 ) . Obviously that the branching-off solution better (in meaning of the functional σ ) approaches the prescribed function by the module.

6. Conclusion

(a)

(b)

Figure 4. The modulus (а) and argument (b) of approximation function.

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 c1 , c2 entering into the discrete Fourier Transform. As follows from investigations, presented, in particular, in [3,17] (for a special case, when F ( s1 , s2 ) = F1 ( s1 ) ⋅ F2 ( s2 ) ), the quantity of the existing solutions grows considerably with increase of the parameters c1 , c2 . 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 F ( s1 , s2 ) at rather small values of parameters c1 , c2 . 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. 1.2 |f | 0.8 0.6 0.410 0.2 -1 -0.5 0

Figure 5. The optimum amplitude of Fouier transform coefficients. Copyright © 2010 SciRes.

3 12 0.5

1 s2

Figure 6. The given (curve 1) and approximation functions in the section s2 = 0 corresponding to branching-off (curve 2) and real (curve 3) solutions. AM

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When 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 arg f ( s1 , s2 ) it is necessary to choose an initial approxi-

mation arg f 0 ( s1 , s2 ) of the same type of parity according to (42). To obtain the irrefragable answer concerning the branching-off solutions for certain values of parameters c1 , c2 it is necessary to use the branching theory of solutions [18]. It is the object of special investigations.

7. References [1]

B. M. Minkovich and V. P. Jakovlev, “Theory of Synthesis of Antennas, ” Soviet Radio, Moscow, 1969.

[2]

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.

[3]

P. O. Savenko, “Nonlinear Problems of Radiating Systems Synthesis (Theory and Methods of the Solution),” Institute for Applied Problems in Mechanics and Mathematics, Lviv, 2002.

[4]

G. M. Vainikko, “Analysis of Discretized Methods,” Таrtus Gos. University of Tartu, Tartu, 1976.

[5]

R. D. Gregorieff and H. Jeggle, “Approximation von Eigevwertproblemen bei nichtlinearer Parameterabhängikeit,” Manuscript Math, Vol. 10, No. 3, 1973, pp. 245271.

[6]

[7]

O. Karma, “Approximation in Eigenvalue Problems for Holomorphic Fredholm Operator Functions I,” Numerical Functional Analysis and Optimization, Vol. 17, No. 3-4, 1996, pp. 365-387. M. A. Aslanian and S. V. Kartyshev, “Updating of One Numerous Method of Solution of a Nonlinear Spectral Problem,” Journal of Computational Mathematics and Mathematical Physics, Vol. 37, No. 5, 1998, pp. 713-717.

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[8]

S. I. Solov’yev, “Preconditioned Iterative Methods for a Class of Nonlinear Eigenvalue Problems,” Linear Algebra and its Applications, Vol. 41, No. 1, 2006, pp. 210-229.

[9]

P. A. Savenko and L. P. Protsakh, “Implicit Function Method in Solving a Two-dimensional Nonlinear Spectral Problem,” Russian Mathematics (Izv. VUZ), Vol. 51, No. 11, 2007, pp. 40-43.

[10] V. A. Trenogin, “Functional Analysis,” Nauka, Moscow ,1980. [11] I. I. Privalov, “Introduction to the Theory of Functions of Complex Variables,” Nauka, Moscow, 1984. [12] A. N. Kolmogorov and S. V. Fomin, “Elements of Functions Theory and Functional Analysis,” Nauka, Moscow, 1968. [13] P. P. Zabreiko, А. I. Koshelev and М. А. Krasnoselskii, “Integral Equations,” Nauka, Moscow, 1968. [14] М. А. Krasnoselskii, G. М. Vainikko, and P. P. Zabreiko, “Approximate Solution of Operational Equations,” Nauka, Moscow, 1969. [15] I. I. Liashko, V. F. Yemelianow and A. K. Boyarchuk, “Bases of Classical and Modern Mathematical Analysis,” Vysshaya Shkola Publishres, Kyiv, 1988. [16] E. Zeidler, “Nonlinear Functional Analysis and Its Applications I: Fixed-Points Theorem,” Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985. [17] P. A. Savenko, “Synthesis of Linear Antenna Arrays by Given Amplitude Directivity Pattern,” Izv. Vysch. uch. zaved. Radiophysics, Vol. 22, No. 12, 1979, pp. 1498-1504. [18] М. M. Vainberg and V. А. Trenogin, “Theory of Branching of Solutions of Nonlinear Equations,” Nauka, Moscow, 1969. [19] V. V. Voyevodin and Y. J. Kuznetsov, “Matrices and Calculations,” Nauka, Moscow, 1984. [20] A. Gursa, “Course of Mathematical Analysis, Vol. 1, Part 1,” Moscow-Leningrad, Gos. Technical Theory Izdat, 1933. [21] V. I. Smirnov, “Course of High Mathematics, Vol. 1,” Nauka, Moscow, 1965.

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