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
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 [9], 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 [9]. 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
( )
22
11
()
121122
()
(,)exp
NMn
nmnmnm
nNmMn
fssIicxscys
=−=−

=+

∑∑ %%
setting here
nmx
xn
=∆
()
nNN
=−÷ ,
nmy
ym
=∆
()
mMM
=−÷ ; 11
x
cc
=∆
%
, 22
y
cc
=∆
%
. If it is necessary
for the accepted assumptions we shall consider the for-
mula
( )
121122
(,)exp
NM
nm
nNmM
fssAIicnscms
=−=−
=≡+


∑∑
I (1)
as a linear operator, acting from complex finite-dimensional
space
22
NM
I
H× =
£
(
2
21
NN
=+
,
2
21
MM
=+
) into
the space of complex-valued continuous functions with
respect to two real variables determined in the domain
(
)
{
}
121122
,:,
ssscsc
Ω=≤π≤π .

P. SAVENKO ET AL. 66
Here
1
c
,
2
c
are any real non-dimensional numerical
parameters belonging to
(
)
{
}
1212
,:0,0
c
cccacb
Λ=<≤<≤
.
The function
12
(,)
fss
is
1
2
c
π
- periodic function
on argument
1
s
and
2
2
c
π - periodic on
2
s
.
In considered spaces we introduce scalar products and
generable by them norms
( )
2
12 12
4
,I
NM
nmnm
HnNmM
II
cc =−=−
π
=∑∑ II ,
(
)
1/2
,
I
H
=
III
, (2)
(
)
( )
(
)
(
)
(2)
1211221212
,,,
C
fffssfssdsds
Ω
Ω
=∫∫ ,
(
)
( )
(2)
1/2
,C fff
Ω
=. (3)
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
( )
2
222 1212 ,
,nm
nm
AfssdsdsI
II
Ω
===
∑
∫∫ (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
( )()
12
12112212
2,exp
4
cc
Affssicnscmsdsds
∗
Ω
=−+

π
∫∫
(
)
,
nNNmMM
=−÷=−÷ . (5)
Let such function be given
1212
12
12
(,),(,),
(,)
0,(,)\,
FssssG
Fss
ssG

∈⊆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
() FAFf
CC
II
ΩΩ
σ=−≡− (7)
in the Hilbertian space
I
H
. Taking into account (4) and
(5), we write the functional
()
I
σ
in a simplified form
(
)
( )
(
)
(2) (2)
2
22
()2,
I
H
FFA
CC
III
Ω
Ω
σ=−+
. (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
I
H
that are represented in the vector and expanded forms,
respectively:
( )
{
}
expargAFiA
II
∗
=


, (9)
( )
12 12
2,exparg
4
NM
nmnm
kNlM
cc
IFssiI
=−=−
Ω


=×
 
π

∑∑
∫∫
( )
)
( )
}
1122112212
exp
icksclscnscmsdsds

×+−+ 
 
(
)
,
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()
fQfKQQFQifQdQ
Bc
Ω
′′′′
=≡
∫∫ , (10)
where
12
(,)
Qss
′′′
=,
12
dQdsds
′′′
=,
(
)
12
,
cc
c=;
(
)
(
)
(
)
11112222
,,,,,,
KQQKsscKssc
c
′′′
=⋅ , (11)
()( )
1
1111111
,,exp
2
N
nN
c
Ksscicnss
=−
′′
=−≡


π∑
( )
( )
21 11
1
1
11
sin 2
2
2
Nc
ss
ccss

′
−


≡π′
−,
()( )
2
2222222
,,exp
2
M
mM
c
Ksscicmss
=−
′′
=−≡


π∑
( )
( )
22 22
2
2
22
sin 2
2
2
Mc ss
cc
ss

′
−


≡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
()0
fQ
′
→
. Ob-
viously that the function
( )
(
)
12
22
()()()
exparg() ()
()()
fQuQivQ
ifQ fQ uQvQ
′′′ +
′
=≡
′′′
+
is continuous if
()Re()
uQfQ
′′
= and
()Im()
vQfQ
′′
=
()Im()vQfQ
′′
=
are continuous functions, where
(
exparg()1
ifQ
)
exparg()1
ifQ
′
=
for any
()
fQ
′
. If
()0
uQ
′
→
and
()0
vQ
′
→
simultaneously then
()0
fQ
′
→
is a complex zero. Its
argument is undetermined accordingly to definition of

P. SAVENKO ET AL. 67
complex zero [11, p. 20]. On this basis we redefine
(
)
exparg()
ifQ
′
at
()0
uQ
′
→
and
()0
vQ
′
→
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
fA
I
∗∗
= is the solution of (10); on the contrary, if
f
∗
is the solution of (10) then
[
]
{
}
exparg()
AFif
I
∗
∗∗
= (12)
is the solution of (9).
Proof. Let
I
∗
be a solution of (9). Then
AFiA
∗
∗∗
−≡
II
( )
{
}
exparg0
AFiA
∗
∗∗
−≡


II
. Acting on this identity by the
linear operator
A
, we have
(
{
exparg0
AAAFiA
II
∗
∗∗
−≡


(
)
}
exparg0
AAAFiA
II
∗∗
−≡


. Since the operator
A
acts from the space
22
NM
I
H× =
£
into the space
(2)
()
C
Ω
and accordingly into
the space 2
()
f
HL
=Ω
%
, and the 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
()
f
HL
∗∗
=Ω
%
into the space
22
NM
I
H× =
£
[10] and the Hilbertian
space
2
L
∗
coincides with the space
2
L
[10]. From here
follows, that
A
∗
acts from the space 2
()
f
HL
=Ω
%
into
the space
22
NM
I
H× =
£
. 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()
f
FifH
∗∈.
Thus
(
)
(
)
exparg()
I
AFifH
∗∗∗
=∈I and the right part
of (10) is the result of action of operator
A
on an ele-
ment
I
∗
, i.e.
(
)
(
)
exparg()
AAAFiff
I∗
∗∗∗
==
. Writ-
ing this equality as
( )
(
)
(
)
exparg()0
AAFiAII
∗
∗∗
−=
and taking into account that a set of operator nulls con-
sists of only a null element we obtain
(
)
(
)
exparg()
AFiAII
∗
∗∗
=. So,
(
(
exparg()
AFif
I∗
∗∗
=
)
)
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
(
)
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).
2) If function
()
fQ
is a solution of (10), then
(
)
exp()
ifQ
β 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
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 unique-
ness of solutions we set the parameter
0
β=
.
Consider the operator
(
)
,,()
DfKQQfQdQ
c
Ω
′′′
≡∫∫ (13)
and corresponding to it quadratic form
(
)
,(,,)()()
DffKQQfQdQfQdQ
c
ΩΩ
′′′
==
∫∫∫∫
( )
2
11221212
exp(,)
NM
nNmM
icnscmsfssdsds
=−=− Ω
=+=


∑∑∫∫
2
12
22
0
cc

ππ
=≥


I.
Obviously that this inequality modifies into equality
only as
0
I
=
. From here follows that the kernel
(
)
,,
KQQ
c
′
is positively defined [13]. Accordingly op-
erator
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 con-
sider as a direct sum of two real spaces of continuous
functions
()()()
CCC
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:
()
max()
CQ
uuQ
Ω∈Ω
=, ()
max()
CQ
vvQ
Ω∈Ω
=,

P. SAVENKO ET AL. 68
(
)
()()()
max,
CC
fuv
CΩΩΩ
=.
The Equation (10) in the decomplexified space
()
C
Ω
we reduce to equivalent to it system of the nonlinear eq-
uations
( )
122
()
()(,),,() ()()
uQ
uQBuvKQQFQdQ
uQvQ
c
Ω
′
′′′
=≡ ′′
+
∫∫ ,
( )
222
()
()(,),,() ()()
vQ
vQBuvKQQFQdQ
uQvQ
c
Ω
′
′′′
=≡ ′′
+
∫∫ .
(14)
Denote the closed convex set of continuous functions
as
()
R
SC
⊂Ω
supposing that
uv
RRR
SSS
=⊕,
{
}
()
:
u
RC
SuuR
Ω
=≤,
{
}
()
:
v
RC
SvvR
Ω
=≤,
max(,,)()
Q
RKQQFQdQ
c
∈Ω Ω
′′′
=∫∫ .
Theorem 1. The operator 12
(,)
T
BBB= determined
by the Formula (14) maps a closed convex set
R
S
of the
Banach space
()
C
Ω
in itself and it is completely con-
tinuous.
Proof. At first we show that
:()()
BCC
Ω→Ω
. Let
(,)
T
fuv
= be any function belonging to
()
C
Ω
. At
(
)
12
,
c
cc
∈Λ
the kernel
(
)
,,
KQQ
c
′
is a continuous
function with respect to its arguments in the closed do-
main
Ω×Ω
. Then accordingly to the Cantor theorem
[15]
(
)
,,
KQQ
c
′
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
,,,,KQQKQQ
a
ε
′′
−<
cc
, where
aFQdQ
=
()
aFQdQ
Ω
′′
=
∫∫ . On this basis we obtain
() ()
1212
()()(),,,,uQuQFQKQQKQQcc
Ω
′′′
−=−×


∫∫
22
() ()
()() G
uQ dQFQdQ
a
uQvQ
′ε
′′′
×≤=ε
′′
+∫∫ , (15)
since 22
()
max1
()()
Q
uQ
uQvQ
′
∈Ω
′
≤
′′
+.
Analogously we have that 12 ()()vQvQ
−≤ε
when-
ever
(
)
(
)
1122
,,QQQQ
′′
−<δ
, i.e.
(,)()
T
uv C
∈Ω
and
:()()
BCC
Ω→Ω
.
To prove the property of a complete continuity of the
operator 12
(,)
T
BBB= it is necessary to prove its com-
pactness and continuity [12]. Show a continuity
12
(,)
T
BBB=. Let 111 (,)
T
R
fuvS
=∈ be any fixed
function and 222
(,)
T
fuv
= be any function belonging
to
R
S
. It is necessary to show that 12
()
0
ff
C
BB Ω
−→
as 12
()
0
ff
CΩ
−→
. Set 21
uuu
=+∆
, 21
vvv
=+∆
.
Taking into account these equalities we obtain
21
2222
22
2211
11 22
11
22
1
uuu
uvuuvvuv
uv uv
+∆
=
+∆+∆+∆+∆
++ +
.
At ()
0
C
uΩ
∆→
, ()
0
C
vΩ
∆→
we have
12
2222
0, 1122
0()
()()
lim ()()()()
u
v
uQuQ
uQvQuQvQC
∆→
∆→ Ω
−≤
++
()
1
22
0, 11
11
0
() 1
limmax1
(),()
()()
u Q
v
uQ
PuQvQ
uQvQ
∆→ ∈Ω
∆→



≤−+



+



( )
22
1111
()
0
()()(),()
uQ
uQvQPuQvQ

∆
+=

+

, (16)
where
(
)
11
(),()
PuQvQ
=
22
11
22
11
2()()2()()()()
1()()
uQuQvQvQuQvQ
uQvQ
∆+∆+∆+∆
=+ +,
since
(
)
11
0,
0
limmax(),()1
uQ
v
PuQvQ
∆→ ∈Ω
∆→
=
.
Similarly we obtain
12
2222
0, 1122
0
()()
limmax0
()()()()
uQ
v
vQvQ
uQvQuQvQ
∆→ ∈Ω
∆→
−=
++
. (17)
Thus, from (16) and (17) follows
111122 ()
0,
0
lim(,)(,)
u
v
BuvBuv Ω
∆→
∆→
−=
C
( )
0,
0
limmax(),,
u Q
v
FQKQQ
∆→ ∈Ω Ω
∆→
′′
=×
∫∫
c
12
2222
1122
()()
0
()()()()
uQuQ dQ
uQvQuQvdQQ

′′

′
×−=

′′′′′
++

.
Analogously

P. SAVENKO ET AL. 69
()
()
211222 ()
0,
0
lim(,)(,)0
C
C
u
v
BuvBuv C
Ω
Ω
Ω
∆→
∆→
−=
.
So, 12
(,)
T
BBB= is continuous operator from
()
C
Ω
into
()
C
Ω
.
We show that a set of functions
gR
SS
B
= satisfies
conditions of the Arzela theorem [12], i.e. we show that
functions of the set
g
S
are uniformly bounded and
equipotentially continuous. Furthermore
RR
SS
B⊂. Let
( )()
12
,(,),(,)
TT
gwfBuvBuv
=ω=≡B, where
fuv
=
(,)
T
fuv
=
is any function of the set
R
S
. Then as
(
)
(
)
12
,,QQQQ
′′
−<δ
analogously with (15) we have
12
12
()
()()
()() ()
FQdQ
a
wQwQ
QQ FQdQ
a
Ω
Ω
ε

′′

−
ε


≤=


 
εε
ω−ω


′′



∫∫
∫∫ .
Thus functions of the set
gR
SS
B
= are equipoten-
tially continuous.
The uniform boundedness of the set
gR
SS
B
= fol-
lows from an inequality
{
( )
() maxmax(),,
Q
gFQKQQ
Ω∈Ω Ω
′′
=×
∫∫
C
c
22
()
()()
uQ dQ
uQvQ
′
′
×≤
′′
+
( )
22
()
max(),, ()()
Q
vQ
FQKQQdQR
uQvQ
c
∈Ω Ω

′
′′′
≤≤

′′
+

∫∫ ,
where
(,)
T
fuv
= is any function of the set
R
S
and
( )
12
(,),(,)
T
gfBuvBuv
B=≡ . From the last inequality
we have also
RR
SS
B⊂. So, the operator 12
(,)
T
BBB=
is completely continuous mapping a closed convex set
(
)
R
SC
⊂Ω
into itself.
Theorem is proved.
From the Theorem 1 follows satisfaction of conditions
of the Schauder principle [16] according to which the
operator 12
(,)
T
BBB= 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 solu-
tion 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 branch-
ing lines of solutions exist and a problem on finding the
lines of branching is a nonlinear two-parametrical spec-
tral problem.
Easily to be convinced that function
(
)
0(,)(),,
G
fQFQKQQdQ
cc
′′′
=∫∫ (18)
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), branching-off from real solution 0
(,)
fQ
c
, we
consider a problem on finding such set of values of pa-
rameters
(
)
(0)(0)(0)
12
,ccc= and all distinct from
0
(,)
fQ
c
solutions of the system (14) which for
(0)
0
cc
−→
(where
(0)
11
cc
≥,
(0)
22
cc
≥) satisfy con-
ditions
( )
(
)
(0)
max,,0
QGuQfQcc ∈
−→
,
(
)
max,0
QG vQc ∈
→
. (19)
These conditions indicate the need to find small con-
tinuous in
G
solutions
(
)
(
)
(
)
(0)
0
,,,wQuQfQ=−ccc
,
(
)
(
)
,,
QvQω=
cc
,
which converge uniformly to zero as
(0)
cc
→.
Set
(0)
11
cc
=+µ
,
(0)
22
cc
=+ν
(20)
and desired solutions we find in the form
( )
(
)
( )
(0)
0
,,,,
uQfQwQ
=+µν
cc ,
(
)
(
)
,,,
vQQ
=ωµν
c.
(21)
Further we omit dependence of the functions
(
)
,,
wQ
µν
and
(
)
,,
Q
ωµν
on parameters
µ
and
ν
.
Notice the properties of integrand in the system (14).
They are continuous functions with respect to the argu-
ments. After substitution (20) and (21) into (14) the inte-
grand develop in equiconvergent power series by func-
tional arguments
w
and
ω
, numerical parameters
µ
and
ν
in the vicinity of a point
(
)
(
)
(0)(0)
0
,,,0
fQcc:
( )
22
()
(),,
()()
uQ
FQKQQ uQvQ
′
′′
=
′′
+
c

P. SAVENKO ET AL. 70
(
)
(0)
0
,,()()
mnpq
mnpq
mnpq
AQQwQQ
+++≥
′′′
=ωµν
∑c,
( )
22
()
(),,
()()
vQ
FQKQQ uQvQ
′
′′
=
′′
+
c
(
)
(0)
1
,,()()
mnpq
mnpq
mnpq
BQQwQQ
+++≥
′′′
=ωµν
∑c. (22)
Here
(
)
(0)
,,
mnpq
AQQ
c
′,
(
)
(0)
,,
mnpq
BQQ c
′ are coeffi-
cients of expansion continuously dependent on the ar-
guments. Substituting (20) and (22) into (14) and taking
into account that
(
)
(0)
0,fQ
c
′ solves the system (14) we
obtain a system of nonlinear equations with respect to
small solutions
w
,
ω
:
(
)
(
)
(0)(0)
1001
(),,uQaQaQ
=µ+ν+
cc
(
)
(0)
2
,,()()
pqmn
mnpq
mnpq
AQQwQQdQ
+++≥ Ω
′′′′
+µνω
∑
∫∫ c,
(23)
() ( )
(0)
(0)
0
()
()(),, ,
Q
QFQKQQdQ
fQ
Ω
′
ω
′′
ω−=
′
∫∫ cc
(
)
(0)
2
,,()()
pqmn
mnpq
mnpq
BQQwQQdQ
+++≥ Ω
′′′′
=µνω
∑
∫∫ c,
(24)
where
(
)
(
)
(0)(0)
100010
,,,
G
aQAQQdQ
cc
′′
=∫∫ ,
(
)
(
)
(0)(0)
010001
,,,
G
aQAQQdQ
cc
′′
=∫∫ .
3. Nonlinear Two-Parametric Spectral
Problem
For further application of methods of the branching
theory of solutions of nonlinear equations [18] to a sys-
tem (23) and (24) it is necessary to find solutions of dis-
tinct from trivial of the linear homogeneous integral equ-
ation obtained equating to zero the left part of (24)
( )
1212
012
()
()(,),,,()
(,,)
G
FQ
QTccKQQccQdQ
fQcc
′
′′′
ϕ=ϕ≡ϕ
′
∫∫
(25)
under condition
0
(,)0
fQ
c
′
>
. Indicate that the operator
():()()
T
cCC
Ω→Ω
is completely continuous. Proof of
this property is similar to the proof of a complete conti-
nuity of the operator
11
(,)
T
BB
B= 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 nonli-
near 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- 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
,
c
cc
∈Λ
at which (25) has distinct from
identical zero solutions.
In operational form a nonlinear two-parametric prob-
lem is presented as
(
)
1212
(,)(,)0
ccxETccx
≡−=
A. (26)
Here
E
is an identical operator and
12
(,)
Tcc
is a
linear integrated operator acting in the Banach space
()
C
Ω
. It is necessary to find eigenvalue
(
)
(0)(0)
12
,
c
ccc=∈
Λ
and corresponding eigenvectors
(
)
(0)
xC
∈Ω
((0)
0
x
≠
) such that (0)(0)(0)
12
(,)0
ccx
=
A.
By direct check we ascertain that for any values of
parameters
(
)
12
,
c
cc ∈
Λ
the function
(
)
0
ˆ(,)(),,
QFQKQQdQ
Ω
′′′
ϕ=
∫∫
cc
(27)
is one of eigenfunctions.
Write a conjugate to (25) equation required in later
( )
0
()
()(),,()
(,)
FQ
QTKQQQdQ
fQ
∗
Ω
′′′
ψ=ψ≡ψ
∫∫
cc
c. (28)
At arbitrary
(
)
12
,
c
cc ∈
Λ
the function
0
ˆ
()()
QFQ
ψ= . (29)
is one of eigenfunctions of (28)
The existence of distinct from identical zero solutions
of (25) at arbitrary
(
)
12
,
c
cc
∈
Λ
testifies to the exis-
tence of coherent components of a spectrum contermin-
ous with the domain
c
Λ
.
For finding the distinct from 0
ˆ
(,)
Qϕ
c
solutions we
exclude from the kernel of integral Equation (25) the
eigen function (27), namely: consider the equation
(
)
(,)(),,()
QTQQQdQ
ccc
Ω
′′′
ϕ=ϕ≡ϕ
∫∫
%
K, (30)
where
() () ()( )
00
0
()
,,,,(),
,
FQ
QQKQQQQ
fQ
ccc
c
′
′′′
=−ψϕ
′
K,
(31)
2
0
00
ˆ
()
() ˆ
L
Q
Q
ψ
ψ=
ψ
,
( )
(
)
2
0
00
ˆ
,
,ˆL
Q
Q
′
ϕ
′
ϕ=
ϕ
c
c. (32)

P. SAVENKO ET AL. 71
From Schmidt Lemma [18] follows that 0
(,)
Qϕ
c
will not be an eigenfunction of this equation for any val-
ues
(
)
12
,
c
cc ∈
Λ
. Thus from a spectrum of operator
there is excluded coherent component coinciding with
the domain
c
Λ
and corresponding to the function
0
(,)
Qϕ
c
.
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
1
c
,
2
c
. We write (25) as
( )
121122
(,)exp
NM
nm
nNmM
ssxicnscms
=−=−
ϕ=+−


∑∑
0012
(,)
xss
−ψ , (33)
where
nm
x
,
0
x
are constants determined by the formu-
las
( )
1212 1122
01212
(,) exp
22(,,,)
nm ccFss
xicnscms
fsscc
Ω
′′
′′
=−+×


′′
ππ
∫∫
1212
(,)
ssdsds
′′′′
×ϕ
(
)
,
nNNmMM
=−÷=−÷ ,
0012121212
(,,,)(,)
xssccssdsds
Ω
′′′′′′
=ϕϕ
∫∫ .
From the Formula (33) follows, that the function
12
(,)
ss
ϕ will become known, if will be found
nm
x
,
0
x
.
Multiplication of both parts of (33) by
( )
12 1122
01212
(,) exp
(,,,)
Fss
ickscls
fsscc
′′
′′
−+


′′ at
kNN
=−÷
,
lMM
=−÷
and by
012
(,)
ss
′′
ϕ, and integration over
Ω
gives a homogeneous system of the linear algebraic
equations for finding
nm
x
,
0
x
( )
() 12
,
NM kl
klnmnm
nNmM
xaccx
=−=−
=∑∑
,
kNN
lMM
=−÷


=−÷

. (34)
Here
( )()
(
)
(
)
() ( )
12 ()()
121212
012
,
,,,
1,
kl
klkl
nmnmnm
bcc
acctccdcc
dcc


=−

+


,
( )
( )
1212
12 201212
(,)
,
(,,,)
4
kl
nm ccFss
tcc fsscc
Ω
=×
π∫∫
( )( )
{
}
112212
exp
icnkscmlsdsds
×−+−

,
( )
() 1212
12012
201212
(,)
,(,)
(,,,)
4
kl
ccFss
bccss
fsscc
Ω
=ψ×
π∫∫ ,
(
)
112212
exp
icksclsdsds
×−+

,
(
)
(
)
12012112212
,(,)exp
nm
dccssicnscmsdsds
Ω
=ϕ+
 ∫∫ ,
(
)
0120120121212
,(,)(,,,)
dccssssccdsds
Ω
=ψϕ
∫∫ .
For coefficients of the matrix
( )
1111
,,
Mnm
ccacc
A
=
(
)
()
1111
,,
,,
kl
Mnm
knNN
mlMM
ccacc
=−÷
=−÷
=
the equality
(
)
() 11
,
lk
mn
acc
=
( )
()
11
,
kl
nm
acc
= is valid, i.e.
M
A
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
(,)(,)0
MMM
ccccxEAx
≡−=
A, (35)
where
M
E
is a unit matrix of dimension
22
NM
×.
In order that the system (34) should have distinct from
zero solutions, it is necessary
(
)
1212
(,)det(,)0
MM
ccccEA
Ψ=−=
. (36)
It is easy to be convinced, that
12
(,)
cc
Ψ
is a real
function. Really as
12
(,)
M
cc
T
is the Hermitian matrix
then it is obvious that
(
)
12
(,)
M
cc
EA
−
is also the
Hermitian matrix. It is known [19] that the determinant
of the Hermitian matrix is a real number. So,
12
(,)
cc
Ψ
is a real function with respect to the real arguments
1
c
and
2
c
.
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
(,)
cc
Ψ.
Consider a necessary later on auxiliary one-dimen-
sional spectral problem (as a special case of the problem
(35)) on the ray
21
cc
=γ
(
γ
is a real coefficient,
(
)
12
,
c
cc
∈Λ
). Introduce into consideration the matrix -
function
(
)
111
(),
MM
ccc
≡γ
%
AA and connected with it
the one-dimensional spectral problem
(
)
(
)
1111
,(,)0
MMM
cccc
γ=−γ=
xEAx
%
A. (37)
It is easy to be convinced, that from the properties of
coefficients of matrix
12
(,)
M
cc
A follows, that the ma-
trix function
12
(,)
M
cc
A is continuous and differentia-
ble on the variables in any open and limited domain
2
c
Λ⊂Λ⊂
¡
. In other words
12
(,)
M
cc
A
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(,)0
MM
cccc
Ψγ=−γ=
EA . (38)
We denote the spectrums of the problems (35) and (37)
as
()
s
A
and
()
s
%
A
, respectively, and the parameter
domain
1
c
as
{
}
111
:0
c
cca
Λ=<≤
. Then for proper-

P. SAVENKO ET AL. 72
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 12 (,)
c
ccc
=∈Λ
the matrix
2222
12
(,)(,)
NMNM
M
cc ×× ∈
££
AL be the Fredholm oper-
ator with a zero index, the matrix - function
(
)
2222
(,):,
NMNM
c××
⋅⋅Λ→ ££AL be holomorphic in the
domain
c
Λ
and
1
()
c
s
≠Λ
%
A. Moreover, let function
(
)
12
,
cc
Ψ be continuously differentiable in
c
Λ
. 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)
1
Nc
%
A and fi-
nite-dimensional root subspace;
2) Each point
(
)
(0)(0)(0)
11
,
c
cc
=γ∈Λ
c is a point of
spectrum of the matrix - function
12
(,)
λλ
A;
3) If
(
)
2
(0)(0)
12
,0
ccc
′
Ψ≠
then in some vicinity of the
point
(0)
1
c
there is a unique continuous differentiable
function
(
)
221
ccc
= solving the Equation (36), i.e. in
some bicircular domain
( )
{
(0)(0)
012111222
,:,
cccccc
Λ=−<ε−<ε
}
(0)(0)
012111222
,:,
cccccc
Λ=−<ε−<ε
there exists a connected component of
spectrum of the matrix-function
12
(,)
cc
A( where
1
ε
,
2
ε
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 [9]. For satisfaction of conditions of Theo-
rem 1 from [9] it is necessary to show that the matrix -
function
(
)
12
,
cc
A is the Fredholm matrix at
12
(,)
c
cc
∈Λ
. This property follows from the known
equality [19]
(
)
(
)
dimkerdimker
∗
=
AA
.
The existence of connected components of spectrum
of the matrix - function
(
)
12
,
cc
A, under condition of
(
)
2
(0)(0)
12
,0
ccc
′
Ψ≠
, follows from the existence theorem
of implicitly given function [20, 21].
Let
()
1
i
c
be a root of (38). Then
(
)
()()()
121
,
iii
c
ccc
=γ∈Λ
is eigenvalue of the problem (33). Consider the equation
12
(,)0
cc
Ψ=
as a problem on finding the implicitly
given function
221
()
ccc
=
in the vicinity of a point
()
1
i
c
for which the conditions of existence theorem [21] are
satisfied. Hence we have the Cauchy problem
1
2
12
2
112
(,)
(,)
c
c
cc
dc
dccc
′
Ψ
=− ′
Ψ, (39)
(
)
()()()
211
iii
ccc
=γ
. (40)
Solving numerically (39) and (40) in some vicinity of
a point
()
1
i
c
, we find the
i
-th connected component of
spectrum (spectral line) of the matrix - function
12
(,)
M
cc
A.
By found solutions of the Cauchy problem at the fixed
values
(
)
()()
12
,
ii
cc
the eigenfunctions of (25) are deter-
mined through the eigenvectors of the matrix
(
)
()()
12
,
ii
M
cc
A obtained by the known methods. Thus
four-dimensional matrix
M
A
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 [2]:
11
()(,)(,,)()
nnn
uQBuvKQQFQ
+
Ω
′′
=≡×
∫∫
c
22
()
()()
n
nn
uQ
dQ
uQvQ
′
′
×′′
+,
12
()(,)(,,)()
nnn
vQBuvKQQFQ
+
Ω
′′
=≡×
∫∫
c
22
()
()()
n
nn
vQ
dQ
uQvQ
′
′
×′′
+ (
0,1,...
n
=
). (41)
After substituting the function
arg()arctg()()
nnn
fQvQuQ
=
()
arg()arctg()()
nnn
fQvQuQ
()
/
(obtained on the basis of successive approxima-
tions (41)) into (12), we denote the obtained sequence of
function values as
n
I
{}
. For the sequence
n
I
{}
the
Theorem 4.2.1 from [3] is fulfilled. From here follows,
that the sequence
n
I
{}
is a relaxation one for the func-
tional (7) and numerical sequence
()
n
σ
I
{}
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
ij
U
,
k
V
l
of the space
()
Ω
C
. Here the indices
,,,
ijk
l
have

P. SAVENKO ET AL. 73
values 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
,
ijkijijkk
BUVUBUVV
⊂⊂
UU
()()
lll
,
ijkijk
UVUV
⊂B
UU
()
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
(,)cos2sin
Fssss
=ππ
(Figure 1), given in
the domain
(
)
{
}
1212
,:1,1Gssss
=≤≤⊂Ω
, for
22
1111
NM
×=×
and values of parameters 1
1.6
c=
and 2
1.2
c
=
belonging to the ray
21
0.75
cc
=. 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
2
s
functions
12
arg(,)
fss
and the coefficients of transfor-
mation
,
nm
I
(
,
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
21
0.75
cc
=
: the curve 1
Figure 1. The function
()( )
1212
(,)cos2sin
Fssss
=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
2
s
argument
(
)
12
arg,
fss
. From analysis of Figure
3 follows that at the point 1
0.77
c≈ 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
22
CMc
= 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
21
0.75
Cc
∆=∆
.
An amplitude (а) and argument (b) of approximate
function are given in Figure 4 for 1
1.6
c= and
2
1.2
c=. 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. 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 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 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.
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.5 00.5 1
1
2
3
| f |
s
2
Figure 6. The given (curve 1) and approximation functions
in the section
2
0
s
=
corresponding to branching-off
(curve 2) and real (curve 3) solutions.

P. SAVENKO ET AL. 75
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
(
)
12
arg,
fss
it is necessary to choose an initial approxi-
mation
(
)
012
arg,
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 [18]. It is the object of special investigations.
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