Approximation of Urysohn operator with operator polynomials of Stancu type
We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator. In the case of two variables, the integration domain is a "rectangular isosceles triangle". As a special case, Bernstein-type polynomials are obtained....
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| Date: | 2012 |
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Institute of Mathematics, NAS of Ukraine
2012
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508503710367744 |
|---|---|
| author | Demkiv, I. I. Makarov, V. L. Демків, І. І. Макаров, В. Л. |
| author_facet | Demkiv, I. I. Makarov, V. L. Демків, І. І. Макаров, В. Л. |
| author_sort | Demkiv, I. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:30:02Z |
| description | We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator.
In the case of two variables, the integration domain is a "rectangular isosceles triangle". As a special case, Bernstein-type polynomials are obtained.
The Stancu asymptotic formulas for remainders are refined. |
| first_indexed | 2026-03-24T02:26:15Z |
| format | Article |
| fulltext |
© V. L. MAKAROV, I. I. DEMKIV, 2012
318 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
UDC 519.65
V. L. Makarov (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
I. I. Demkiv (Lviv Polytech. Nat. Univ.)
APPROXIMATION OF URYSOHN OPERATOR
WITH OPERATOR POLYNOMIALS OF STANCU TYPE
НАБЛИЖЕННЯ ОПЕРАТОРА УРИСОНА
ОПЕРАТОРНИМИ ПОЛІНОМАМИ ТИПУ СТАНКУ
We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn
operator. In the case of two variables, the integration domain is a “rectangular isosceles triangle”. As a special case, Bern-
stein-type polynomials are obtained. The Stancu asymptotic formulas for remainders are refined.
Досліджується однопараметрична сім’я додатних поліноміальних операторів від однієї та двох змінних, що набли-
жають оператор Урисона. У випадку двох змінних областю інтегрування є „прямокутний рівнобедрений три-
кутник”. Як окремий випадок, одержано поліноми типу Бернштейна. Дано уточнення асимптотичних формул
Станку для залишкових членів.
1. Introduction. The problem of one and two variables operator Urysohn approximation with
polynomial operator of Stancu type, when the form of integrand function f is unknown and acces-
sible information is the only information about the smoothness of f and value of operator F on
the given sequence of it arguments. Stancu polynomials are one-parameter generalization of Bern-
stein polynomials, which were studied and continue to being studied by many authors, starting with
the works of D. Stancu [1, 2].
Bernstein polynomials for approximation of Urysohn operator was first applied in the paper [3].
Here, the linear operator relative to the F is suggested and investigated
Bn F, x !( )( ) = F t, 0( ) "
#F t, k
n
H ! " z( )$
%&
'
()
#z
Cnkxk z( ) 1" x z( )[ ]n"k dz
k=0
n
*
0
1
+ ,
that with increasing of its order however precisely approximates Urysohn operator
F t, x !( )( ) = f t, z, x z( )( ) dz
0
1
" (1)
where H !( ) is Heaviside function. Another approach to polynomial approximation, applied on
Lagrange interpolation polynomial, was proposed in paper [4]
In [5] the obtained results have been generalized on the function of two variables, when integra-
tion domain is a rectangle.
The purpose of this work is investigation of linear operator polynomial that approximates
Urysohn operator in the case (1), and also, when integrand function f has the form f (t, z1, z2,
x(z1), y(z2 )), when integration domain is a “rectangular isosceles triangle” !2 = {(z1, z2 ) : z1 " 0;
z2 ! 0; z1 + z2 " 1} and Urysohn operator is defined as follows:
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 319
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
F t, x !( ) , y !( )( ) = f t, z1, z2, x z1( ) , y z2( )( ) dz1dz2
"2
## .
As the basis of this polynomial one-parameter flock of D. Stancu polynomials are suggested (see
[1, 2]), and it serves for obtaining Bernstein polynomials as a particular case. The specification of
asymptotic formulas for the D. Stansu remainder terms both in the case of one and two variables is
given. A number of numerical examples is put to illustrate the theory.
2. The case of one variable. Let us consider Urysohn operator (1) with unknown kernel
f t, z, x z( )( ), its properties can be judged only by its influence on any functions x z( ) from cer-
tain class. In technique such situation is sometimes called “grey box”.
The problem lies in the development of simple polynomial approximation of operator F that
with the increasing of its order could approximate F with high accuracy.
Let the continuous interpolation conditions are set
F t, xi !, "( )( ) = f t, z, xi z, "( )( ) dz
0
1
# , i = 0, n ,
where
xi z, !( ) = i
n
H z " !( ) , ! " 0,1[ ], i = 0, n .
It is necessary to develop the approximation to Urysohn operator (1) with unknown function
f t, z, x z( )( ) by means of noted function F t, xi , ! "( )( ) . As the basis of such approximation we
will take a linear positive operator polynomial, investigated in [1], which approximates the function
of one variable, defined and bounded in the interval 0,1[ ]. We have
Pn
![ ] F, x "( )( ) = Cnkvn(k )
k=0
n
# x z( ) ,!( ) f t, z, k
n
$
%&
'
() dz = pn[!]( f (t, z, "); x(z)) dz
0
1
*
0
1
* , (2)
where
vnk( ) x z( ) ,!( ) =
x z( ) + k1!( ) 1" x z( ) + k2!( )k2 =0
n"k"1#k1=0
k"1#
1+ k3!( )k3=0
n"1#
,
! is nonnegative parameter that can depend only on n .
Note, that the node x0 z, !( ) = 0 and continuous node xn z, !( ) = H z " !( ) will be the inter-
polation nodes. If ! = "
1
n
, then all functions xi z, !( ) = i
n
H z " !( ), i = 0,1,…, n , ! "[0,1]
will be continuous interpolation nodes. Really, in this case formula (2) acquires the form
Pn
! 1
n
"
#$
%
&' F, x (( )( ) = Cnkvn(k )
k=0
n
) x z( ) , ! 1
n
*
+,
-
./ f t, z, k
n
*
+,
-
./ dz = pn
! 1
n
"
#$
%
&' ( f (t, z, (); x(z)) dz
0
1
0
0
1
0 .
320 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Let substitute the function xi z, !( ) = i
n
H z " !( ) into it, then we obtain
Pn
! 1
n
"
#$
%
&' F, i
n
H ( ! )( )*
+,
-
./ = Cnkvn(k )
k=0
n
0 0, ! 1
n
*
+,
-
./ f t, z, k
n
*
+,
-
./
0
)
1 dz +
+ Cnkvn(k )
k=0
n
! i
n
, " 1
n
#
$%
&
'( f t, z, k
n
#
$%
&
'( dz
)
1
* =
= f t, z, 0( )
0
!
" dz + f t, z, i
n
#
$%
&
'( dz =
!
1
" F t, i
n
H ) * !( )#
$%
&
'( , i = 0,1,…, n , ! "[0,1].
Here we used the equalities
vn(k )
i
n
, ! 1
n
"
#$
%
&' = (i,k , i, k = 0,1,…, n ,
where !i,k is the symbol of Kronecker. Because of uniqueness the interpolation polynomial
pn
! 1
n
"
#$
%
&' ( f (t, z, (); x(z)) coincides with interpolation Lagrange polynomial.
The following identities have been proved in [1]:
pn[!] 1; x( ) = 1, pn[!] "( ) ; x( ) = x , pn[!] "( )2 ; x( ) =
1
1+ !
x 1# x( )
n
+ x x + !( )$
%&
'
()
. (3)
In (2) the unknown functions f t, z, k
n
!
"#
$
%& , i = 0, n , are involved. To define them we use the
results of work [6]. Then we will have
f t, z, k
n
!
"#
$
%& = '
(F t, k
n
H ) ' z( )!
"#
$
%&
(z
+ f t, z, 0( ). (4)
Let us use (4) and reduce (2) to the form
Pn
![ ] F, x "( )( ) = F t, 0( ) # Cnkvnk
k=0
n
$ x z( ) ,!( )
%F t, k
n
H " # z( )&
'(
)
*+
%z
dz
0
1
, ,
that is used in the concrete calculation, but while proving the convergence theorem we will take into
the account formula (2). Let’s suppose that the condition holds
f (t, z, x) !C([0,1]3). (5)
Further we will need the theorem of P. P. Korovkin [7].
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 321
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Let
Ln ( f ; x) = f (t) d!n (x, t)
a
b
" , n = 1, 2,…,
is the sequence of linear operators, defined for f (t) !C[a, b], where !n (x, t) for every n and for
every fixed x is a function of bounded variation of variable t in [a, b]. Then the following state-
ment is valid:
If in a, b[ ] Ln (t i; x) converges uniformly to t i , i = 0,1, 2 , then Ln ( f ; x) converges uni-
formly to f (x) for any f (t) !C[a, b].
The following theorem is true.
Theorem 1. Let the Urysohn operator (1) is such, that the function f t, z, x( ) satisfies the con-
dition (5), and let the operator (1) is considered in the compact ! " C 0,1[ ] and 0 ! " =
= ! n( )" 0 as n! " , then the sequence of operators Pn! F, x "( )( ){ } converges to
F t, x !( )( ) uniformly with respect to x t( ) !" , where ! = x z( ) "C 0,1[ ]: 0 # x t( ) # 1{ } .
Proof. For every fixed t, z ![0,1] operator polynomial Pn[!]( f (t, z, "); x(z)) from (2), with re-
spect to P. P. Korovkin theorem converges to f (t, z, x(z)) anywhere on the compact ! , if
0 ! " = " m( )# 0 as m! " . Then, from the evident generalization of Theorem 1 [8, p. 506]
(boundary conversion under integral sign. Chapter. Integrals, that depend on the parameter), in our
case the Theorem 1 statement follows.
3. Estimate of the approximation order in case of one variable. Next we will use the
modulus of continuity, defined as follows:
! f , "( ) = ! "( ) = max
t ,z# 0,1[ ]
sup
x$ !x %"
f t, z, x( ) $ f t, z, !x( ) ,
where ! does not depend on x , !x .
Theorem 2. Let Urysohn operator (1) is such, that the function f t, z, x( ) satisfies condition
(5), and let operator (1) is considered on compact ! and ! " 0 , then
F t, x(!)( ) " Pn#[ ] F, x(!)( ) $
3
2
%
1+ #n
n + #n
&
'(
)
*+
. (6)
Proof. Whereas, on the compact ! we have Cnkvnk( ) x z( ) ,!( ) " 0 and Pm![ ] 1, x(")( ) = 1,
then it might be written
F t, x(!)( ) " Pn#[ ] F, x(!)( ) $ f t, z, x z( )( ) " f t, z, k
n
%
&'
(
)*k=0
n
+ Cnkvnk( ) x z( ) ,#( )
0
1
, dz .
Let us use the following module continuity properties
g x( ) ! g !x( ) " # x ! !x( ) , # $%( ) " 1+ $( )# %( ) , $ > 0 . (7)
Then we obtain
322 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
f t, z, x z( )( ) ! f t, z, k
n
"
#$
%
&' ( ) f ; x z( ) ! k
n
"
#$
%
&'
( 1+ 1
*
x z( ) ! k
n
"
#$
%
&'
) f ; *( ) =
= 1+ 1
!
x z( ) " k
n
#
$%
&
'(
) !( ).
Hence
F t, x(!)( ) " Pn#[ ] F, x(!)( ) $ 1+ 1
%
x z( ) " k
n
&
'(
)
*+
, %( )
k=0
n
- Cnkvnk( ) x z( ) ,#( )
0
1
. dz =
= 1+ 1
!
Cnkvnk( ) x z( ) ,"( )
k=0
n
#
0
1
$ x z( ) % k
n
dz
&
'
(
)
*
+ , !( ). (8)
Now taking into account both Cauchy – Schwarz inequality and identities (3) we write
Cnkvnk( ) x z( ) ,!( ) x z( ) " k
nk=0
n
#
0
1
$ dz % Cnkvnk( ) x z( ) ,!( ) x z( ) " k
nk=0
n
#
2&
'
(
(
)
*
+
+
1/2
dz
0
1
$ =
= x2 z( ) ! 2x z( ) Cnkvnk( ) x z( ) ,"( ) k
nk=0
n
# + Cnkvnk( ) x z( ) ,"( ) k
n
$
%&
'
()
2
k=0
n
#
*
+
,
,
-
.
/
/
1/2
dz
0
1
0 ≤
≤ max
x z( )! 0,1[ ]
x z( ) 1" x z( )( )
n
1+ #n
1+ #
$
%&
'
()
1/2
0
1
* dz +
1
2
1+ #n
n + #n
.
Using this, we write (8) in the form
F t, x(!)( ) " Pn#[ ] F, x(!)( ) $ 1+ 1
2%
1+ #n
n + #n
&
'(
)
*+
, %( ) .
To obtain the statement of Theorem 1, we will choose ! =
1+ "n
n + "n
. So we have proved the
theorem.
In the case when ! = 0 , (6) transforms into the following inequality for the operator of Bern-
stein type:
F t, x(!)( ) " Bn F, x(!)( ) #
3
2
1
n
.
4. Asymptotic estimate of approximation error in case of one variable. Let us determine an
asymptotic estimate for the error
Rn
![ ] F, x(")( ) = F t, x(")( ) # Pn![ ] F, x(")( ).
The following theorem is true.
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 323
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Theorem 3. Let Urysohn operator (1) is such, that the function f t, z, x !( )( ) satisfies condi-
tion (5) and has the second continuous derivative with respect to x ![0,1]. Then by
! = !(n)" 0 when n! " we have asymptotic formula for the remainder
Rn
![ ] F, x(")( ) = #
1
2
1+ !n
1+ !
x z( ) 1# x z( )( )
n0
1
$
%2
%x2
f (t, z, x(z))dz + &n
![ ] x z( )( ) dz
0
1
$ , (9)
where
lim
n!"
max #, #
n
, 1
n
$
%&
'
()
*1
+n
#[ ] x(z)( ) = 0 .
Proof. From (1) and (2) taking into account (3) one can write
Rn
![ ] F, x(")( ) = Cnkvnk
k=0
n
# x z( ) ,!( ) f (t, z, x(z)) $ f t, z, k
n
%
&'
(
)*
+
,-
.
/0
dz
0
1
1 .
Let us substitute Taylor series at the point x z( ) with the remainder in the integral form
f t, z, k
n
!
"#
$
%& = f t, z, x z( )( ) + k
n
' x z( )!
"#
$
%&
(
(x
f t, z, x z( )( ) + k
n
' s!
"#
$
%&
(2
(s2x z( )
k /n
) f t, z, s( )ds
for f t, z, k
n
!
"#
$
%& .
Let us add and subtract !
2
!x2
f t, z, x z( )( ) in the integral error. We obtain
Rn
![ ] F, x(")( ) = # Cnkvnk
k=0
n
$ x z( ) ,!( ) k
n
# x z( )%
&'
(
)*
+
+x
f t, z, x z( )( ) dz
0
1
, –
– Cnkvnk
k=0
n
! x z( ) ,"( ) k
n
# s$
%&
'
()
*2
*s2
f t, z, s( ) # *2
*x2
f t, z, x z( )( )+
,
-
.
/
0
x z( )
k /n
1 ds
+
,
-
-
.
/
0
0
dz
0
1
1 –
– Cnkvnk
k=0
n
! x z( ) ,"( ) k
n
# s$
%&
'
()
*2
*x2
f t, z, x z( )( )
x z( )
k /n
+ ds
,
-
.
.
/
0
1
1
dz
0
1
+ .
Let us use the identities (3). We will get
Rn
![ ] F, x(")( ) = #
1
2
1+ !n
1+ !
x z( ) 1# x z( )( )
n
$2
$x2
f (t, z, x(z))
%
&
'
0
1
( –
324 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
– Cnkvn(k ) x z( ) ,!( )
k=0
n
" #2
#s2
f (t, z, s) $ #2
#x2
f (t, z, x(z))
%
&'
(
)*
k
n
$ s%
&'
(
)* ds
+
,
-
-
dz
x(z)
k /n
. =
= ! 1
2
1+ "n
1+ "
x z( ) 1! x z( )( )
n0
1
#
$2
$x2
f (t, z, x(z))dz + %n
"[ ] x z( )( ) dz
0
1
# .
Now, let us estimate !n"[ ] x z( )( ) :
!n
"[ ] x z( )( ) # Cnkvn(k ) x z( ) ,"( )
k=0
n
$ %
&2
&x2
f (t, z, x); s ' x(z)
(
)*
+
,-
k
n
' s ds
x(z)
k /n
. ≤
≤ Cnkvn(k ) x z( ) ,!( )
k=0
n
" 1+ 1
#
s $ x(z)%
&'
(
)* +
,2
,x2
f (t, z, x); #
%
&'
(
)*
k
n
$ s ds
x(z)
k /n
- ≤
≤ Cnkvn(k ) x z( ) ,!( )
k=0
n
" 1+ 1
#
s $ x(z)%
&'
(
)*
k
n
$ s ds
x(z)
k /n
+ ,
-2
-x2
f ; #
%
&'
(
)*
≤
≤ Cnkvn(k ) x z( ) ,!( )
k=0
n
" 1
2
k
n
# x(z)$
%&
'
()
2
+
1
6*
k
n
# x(z)
3$
%
&
'
(
) +
,2
,x2
f ; *
$
%&
'
()
≤
≤ Cnkvn(k ) x z( ) ,!( )
k=0
n
" k
n
# x(z)$
%&
'
()
4*
+
,
-,
.
/
,
0,
1/2
×
× Cnkvn(k ) x z( ) ,!( )
k=0
n
" 1
2
+
1
6#
k
n
$ x(z)%
&'
(
)*
2+
,
-
.-
/
0
-
1-
1/2
2
32
3x2
f ; #
%
&'
(
)*
≤
≤ Cnkvn(k ) x z( ) ,!( )
k=0
n
" k
n
# x(z)$
%&
'
()
4*
+
,
-,
.
/
,
0,
1/2
×
× Cnkvn(k ) x z( ) ,!( )
k=0
n
" 1
2
+
1
18#2
k
n
$ x(z)
2%
&
'
(
)
*
+
,
-
.-
/
0
-
1-
1/2
2
32
3x2
f ; #
%
&'
(
)*
.
By means of Maple’s computer algebra we obtain
Cnkvn(k ) x,!( )
k=0
n
" k
n
# x$
%&
'
()
4
=
= x 1! x( )
1+ 3"( ) 1+ 2"( ) 1+ "( ) x 1! x( ) + 2" 3x2 ! 3x + 1( )#
$
%
&
'
() 3"2 + 6"
n
'
()
*
+, +
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 325
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
+ 1
n2
3x 1! x( ) + " 24x2 ! 24x + 7 ! "( )#
$
%
& +
1
n3
6x2 ! 6x + 1! "( ) '() (10)
and, besides
Cnkvn(k ) x z( ) ,!( )
k=0
n
" 1
2
+
1
18#2
k
n
$ x(z)
2%
&
'
(
)
* =
= 1
2
+
1
18!2
x(z)(1" x(z))
n
1+ #n
1+ #
$
1
2
+
1
72!2
1
n
1+ #n
1+ #
.
Choosing ! =
1+ "n
n + "n
and taking into account stated above, we have
!n
"[ ] x z( )( ) #
1
2
37
72
1
4
+ 2"$
%&
'
() 3"2 + 6"
n
$
%&
'
()
$
%&
+
+ 1
n2
3
4
+ 7!"
#$
%
&' +
1
n3
%
&'
1/2
(
)2
)x2
f ; 1+ !n
n + !n
"
#$
%
&'
. (11)
From here the proof of the theorem follows.
Remark 1. In case ! = 0 the inequality (11) obtains the following form:
!n
0[ ] x z( )( ) "
1
2n
37
72
3
4
+
1
n
#
$%
&
'(
1/2
)
*2
*x2
f ; 1
n
#
$%
&
'(
,
which corresponds to the Theorem 7 from [9] at ! = 0 . But when ! > 0 this theorem most likely
is incorrect, though inequality (11) is just an upper estimate. To testify this statement we refer to
Remark 1 from [10] and, besides, let us give one example. Let ! = n"1/3, x = 0.5 , f (t, z, x) =
= x2+1/4 . The result of the calculations done with Maple’s computer algebra is given in the Table 1.
Table 1. Results of calculations.
n 10 100 1000 10000 100000
n !n[n
"1/3 ](0.5) 0. 01502913 0. 03917958 0. 10510528 0. 25982117 0. 6068217
From this table one can see that with n increasing the expression n !n[n
"1/3 ](0.5) also increases.
Remark 2. If ! = "
1
n
, then, as it has been noticed previously, the polynomial
pn[!] f t, z, "( ) ; x z( )( ) coincides with Lagrange interpolation polynomial. Thus, under condition
!n+1
!xn+1
f t, z, x( ) "C 0,1[ ]3( ), the following formula is valid:
326 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Pn
! 1
n
"
#$
%
&' F, x (( )( ) = F t, x (( )( ) + 1
(n + 1)!
x z( ) ! k
n
)
*+
,
-.
/n+1
/xn+1
f t, z, 0 x z( )( )( )
k=0
n
1
0
1
2 dz , (12)
where ! x z( )( ) " 0,1( ) is some intermediate point.
Example 1. Let’s consider Urysohn operator
F t, x !( )( ) = sin z2x z( )( ) dz
0
1
" , x z( ) !" .
and construct operator polynomial (2) for it
Pn![ ] F, x "( )( ) = Cnk
x z( ) + k1!( ) 1# x z( ) + k2!( )k2 =0
n#k#1$k1=0
k#1$
1+ k3!( )k3=0
n#1$
sin
k=0
n
% z2
k
n
&
'(
)
*+ dz
0
1
, .
For instance, let’s choose, ! =
1
n
, ! = "
1
n
and x z( ) = 1! z2
1+ z2
. For calculations we’ll use
Maple. Let’s set Digits:=256. The results we write into Table 2, where
!1 = F t, x(")( ) # Pn1/n[ ] F, x(")( ) , !2 = F t, x(")( ) # Bn F, x(")( ) ,
!3 = F t, x(")( ) # Pn #1/n[ ] F, x(")( ) .
Table 2. Results of calculations of Example 1.
n !1 C1 = n * !1 !2 C2 = n * !2 !3
4 0.00359457 0.0143783 0.00229443 0.0091777 3.9117323e-7
8 0.00203442 0.016275 0.0011621 0.009296 8.4629711e-14
16 0.0010908 0.0174528 0.00058493 0.0093589 6.0363698e-29
32 0.000566202 0.0181185 0.000293464 0.0093909 2.1917865e-63
64 0.00028866 0.0184741 0.000146985 0.009407 3.4561993e-141
We see, that the following inequalities hold:
F t, x(!)( ) " Pn
1
n
#
$%
&
'( F, x(!)( ) )
0.019
n
, F t, x(!)( ) " Bn F, x(!)( ) #
0.095
n
.
Let’s use formula (12). Then, using the reasoning from [11, p. 95] it is easy to convince that the
following estimate is valid:
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 327
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F t, x(!)( ) " Pn
" 1
n
#
$%
&
'( F, x(!)( ) )
1
(n + 1)!
max
x*[0,1]
x "
k
nk=0
n
+ )
1
(n + 1)nn+1
.
Its right-hand side tends to zero very fast. Thus, if n = 4 it is already less than 1
5 ! 45
=
= 10!30.1953125….
5. The case of two variables. Consider Urysohn operator in case of two variables
F t, x !( ) , y !( )( ) = f t, z1, z2, x z1( ) , y z2( )( ) dz1dz
"2
## (13)
with unknown kernel f t, z1, z2, x z1( ) , y z2( )( ) and integration domain
!2 = z1, z2( ) : z1 " 0; z2 " 0; z1 + z2 # 1{ } .
Besides, let the following inequalities hold x z1( ) ! 0, y z2( ) ! 0, x z1( ) + y z2( ) " 1.
Let the interpolation conditions are set
F t, xi !( ) , y j !( )( ) = f t, z1, z2, xi z1( ) , y j z2( )( ) dz1dz
"2
## ,
where
xi z1, !1( ) =
i
m
H z1 " !1( ) , !1 " 0,1[ ], i = 0,m ,
y j z2, !2( ) = j
m
H z2 " !2( ), !2 " 0,1[ ], j = 0,m ,
H !( ) is a Heaviside function, !1 + !2 " 1, 0 ! xi z1, "1( ) + y j z2, "2( ) ! 1, i + j ! m .
The problem lies in the development of operator polynomial approximation to operator (13) by
means of defined function F t, xi !( ) , y j !( )( ) which can approximate F t, x !( ) , y !( )( ) as pre-
cisely as possible with the increasing of its order. As the bases we’ll take a linear positive polyno-
mial operator from [2], which approximates the two variable function f x, y( ), that is bounded in
the “rectangular isosceles triangle”
! = x, y( ) : x " 0; y " 0; x + y # 1{ }.
We gain the next operator polynomial
Pm![ ] F, x "( ) , y "( )( ) = wmi, j( ) x z1( ) , y z2( ) ,!( ) f t, z1, z2,
i
m
, j
m
#
$%
&
'(0)i+ j)m
*
+2
,, dz1dz2 =
= pm[!](t, z1, z2, x("), y("))
#2
$$ dz1dz2 , (14)
328 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
where
wmi, j( ) x z1( ) , y z2( ) ,!( ) = Ci, j m( ) vmi, j( ) x z1( ) , y z2( ) ,!( ), Ci, j m( ) = m !
i ! j ! m ! i ! j( )! ,
vmi, j( ) x z1( ) , y z2( ) ,!( ) =
=
x z1( ) + k1!( ) y z2( ) + k2!( ) 1" x z1( ) " y z2( ) + k3!( )k3=0
m"i" j"1#k2 =0
j"1#k1=0
i"1#
1+ k4!( )k4 =0
m"1#
,
! is a nonnegative parameter that depends only on m .
Note, that from wmi, j( ) x z1( ) , y z2( ) ,!( ) " 0 in !2 follows, that linear operator (14) is posi-
tive in !2 in the sense, that if f t, z1, z2,
i
m
, j
m
!
"#
$
%& ' 0 in !2 , then Pm![ ] F, x "( ) , y "( )( ) # 0
in !2 .
In the case ! = 0 operator (14) transforms into Bernstein operator in the form
Bm F, x !( ) , y !( )( ) =
= m !
i ! j ! m ! i ! j( )! x
i (z1)y j (z2 ) 1! x(z1) ! y(z2 )( )m!i! j f t, z1, z2,
i
m
, j
m
"
#$
%
&'0(i+ j(m
)
*2
++ dz1dz2 .
Lets show, similarly to one dimensional case, that if ! = "
1
m
the nodes
i
m
H z1 ! "1( ) , j
m
H z2 ! "2( ){ } , 0 # i + j # m, z1, z2{ } $%2 ,
i
m
H z1 ! "1( ) , 0{ } , 0, jm H z2 ! "2( ){ } , z1, z2, "1, "2 #[0,1],
will be continuous interpolation nodes. For the last two nodes this follows from one-dimensional
case. Then we have
Pm
! 1
m
"
#$
%
&' F, k
m
H ( ! )1( ) , l
m
H ( ! )2( )*
+,
-
./ =
= wmi, j( ) k
m
H z1 ! "1( ) , l
m
H z2 ! "2( ) , ! 1
m
#
$%
&
'( f t, z1, z2,
i
m
, j
m
#
$%
&
'(0)i+ j)m
*
+2
,, dz1dz2 =
= wmi, j( ) 0, 0, ! 1
m
"
#$
%
&' f t, z1, z2,
i
m
, j
m
"
#$
%
&'0(i+ j(m
)
0
1!*1
+
0
*1
+ dz2dz1 +
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 329
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
+ wmi, j( ) 0, l
m
, ! 1
m
"
#$
%
&' f t, z1, z2,
i
m
, j
m
"
#$
%
&'0(i+ j(m
)
1!*1
1!z1
+
0
*1
+ dz2dz1 +
+ wmi, j( ) k
m
, l
m
, ! 1
m
"
#$
%
&' f t, z1, z2,
i
m
, j
m
"
#$
%
&'0(i+ j(m
) dz2dz1
0
1!z1
*
+1
1!+2
* +
+ wmi, j( ) k
m
, 0, ! 1
m
"
#$
%
&' f t, z1, z2,
i
m
, j
m
"
#$
%
&'0(i+ j(m
)
0
1!z1
*
1!+2
1
* dz2dz1.
Or, taking into account relation [2] wmi, j( ) k
m
, l
m
, ! 1
m
"
#$
%
&' = (i, k( j,l , i, j, k, l = 0,1,…,m , fi-
nally we obtain
Pm
! 1
m
"
#$
%
&' F, k
m
H ( ! )1( ) , l
m
H ( ! )2( )*
+,
-
./ =
= f t, z1, z2, 0, 0( )
0
1!"1
#
0
"1
# dz2dz1 + f t, z1, z2, 0,
l
m
$
%&
'
() dz2dz1
1!"1
1!z1
#
0
"1
# +
+ f t, z1, z2,
k
m
, l
m
!
"#
$
%& dz2dz1
0
1'z1
(
)1
1')2
( + f t, z1, z2,
k
m
, 0!
"#
$
%&
0
1'z1
(
1')2
1
( dz2dz1 =
= F t, k
m
H ! " #1( ) , l
m
H ! " #2( )$
%&
'
() .
In (14) unknown functions f t, z1, z2,
i
m
, j
m
!
"#
$
%& are involved. To find them by known ones, we
will use the following relations:
F t, i
m
H ! " z1( ) , j
m
H ! " z2( )#
$%
&
'( = f t, )1, )2,
i
m
H )1 " z1( ) , j
m
H )2 " z2( )#
$%
&
'( d)2d)1
*2
++ ,
F t, i
m
H ! " z1( ) , 0#
$%
&
'( = f t, )1, )2, 0, 0( )
0
1")1
* d)2d)1 + f t, )1, )2,
i
m
, 0#
$%
&
'( d)2d)1
0
1")1
*
z1
1
*
0
z1
* ,
F t, 0, j
m
H ! " z2( )#
$%
&
'( = f t, )1, )2, 0, 0( )
0
1")2
* d)1d)2 + f t, )1, )2, 0,
j
m
#
$%
&
'( d)1d)2
0
1")2
*
z2
1
*
0
z2
* ,
from which we can easily obtain
f t, z1, z2,
i
m
, j
m
!
"#
$
%& =
'2
'z1'z2
F t, i
m
H ( ) z1( ) , j
m
H ( ) z2( )!
"#
$
%& + f t, z1, z2,
i
m
, 0!
"#
$
%& +
330 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
+ f t, z1, z2, 0,
j
m
!
"#
$
%& ' f t, z1, z2, 0, 0( ) ,
!
!z1
F t, i
m
H " # z1( ) , 0$
%&
'
() = f t, z1, *2, 0, 0( )
0
1#z1
+ d*2 # f t, z1, *2,
i
m
, 0$
%&
'
() d*2
0
1#z1
+ , (15)
!
!z2
F t, 0, j
m
H " # z2( )$
%&
'
() = f t, *1, z2, 0, 0( )
0
1#z2
+ d*1 # f t, *1, z2, 0,
j
m
$
%&
'
() d*1
0
1#z2
+ .
Let’s substitute (15) into (14) and use the equalities
Ci, j (m)
j=0
m!i
" vmi, j( ) x, y,#( )
i=0
m
" g(i) = Cmi vmi x,#( ) g(i)
i=0
m
" ,
(16)
Ci, j (m)
j=0
m!i
" vmi, j( ) x, y,#( )
i=0
m
" g( j) = Ci, j (m)
i=0
m! j
" vmi, j( ) x, y,#( )
j=0
m
" g( j) = Cmj vmj x,#( ) g( j)
j=0
m
" .
Then operator (14) can be written in the form
Pm
![ ] F, x "( ) , y "( )( ) =
= Ci, j (m)
j=0
m!i
" vmi, j( ) x z1( ) , y z2( ) ,#( )
i=0
m
"
$2
%%
&2
&z1&z2
F t, i
m
H ' ! z1( ) , j
m
H ' ! z2( )(
)*
+
,- dz1dz2 –
– Cmi vmi x z1( ) ,!( ) "
"z1
F t, i
m
H # $ z1( ) , 0%
&'
(
)*i=0
m
+ dz1
0
1
, –
– Cmj vmj y z2( ) ,!( ) "
"z2
F t, 0, j
m
H # $ z2( )%
&'
(
)*j=0
m
+
0
1
, dz2 + F t, 0, 0( ), (17)
where
vmi( ) x z1( ) ,!( ) =
x z1( ) + k1!( ) 1" x z1( ) + k3!( )k3=0
m"i"1#k1=0
i"1#
1+ k4!( )k4 =0
m"1#
,
vmj( ) y z2( ) ,!( ) =
y z2( ) + k1!( ) 1" y z2( ) + k3!( )k3=0
m" j"1#k1=0
j"1#
1+ k4!( )k4 =0
m"1#
.
Formula (17) has a constructive character and it can be used in the practical calculations, but
while theoretical investigating we will use formula (14) instead.
Let’s consider, that
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 331
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
f t, z1, z2, x, y( ) !C 0,1[ ]5( ). (18)
Then we will need the following theorem.
Theorem 4 [2]. If f x, y( )!C(") and 0 ! " = "(m)# 0 , when m! " , then the sequence
of operators pm[!]( f ; x, y){ } uniformly converges to f x, y( ) on ! .
Proving this theorem we used the next properties of operator polynomial pm[!]( f ; x, y) :
pm[!](1; x, y) = 1, pm[!](t; x, y) = x , pm[!]("; x, y) = y ,
pm[!](t 2; x, y) =
1
1+ !
x(1" x)
m
+ x(x + !)#
$%
&
'(
, (19)
pm[!](t"; x, y) = 1# 1
m
$
%&
'
()
xy
1+ !
, pm[!]("2; x, y) =
1
1+ !
y(1# y)
m
+ y(y + !)$
%&
'
()
.
Theorem 5. Let condition (18) holds and operator (13) is considered on the compact ! and
0 ! " = " m( )# 0 if m! " , then the sequence of operators Pm! F, x "( ) , y "( )( ){ } uniformly
converges to F t, x !( ) , y !( )( ) relatively to x z1( ) , y z2( ){ } !" , where ! = x z1( ) "C 0,1[ ]{ ,
y z2( ) !C 0,1[ ], 0 ! x z1( ) + y z2( ) ! 1}.
Proof. For every fixed t, z1, z2 ![0,1] operator polynomial pn[!]( f (t, z1, z2, ", "); x(z1), y(z2 ))
from (14), in conformity with Theorem 4 converges to f t, z1, z2, x(z1), y z2( )( ) uniformly any-
where on the compact if 0 ! " = " m( )# 0 as m! " . Then, from the evident generalization
of Theorem 1 [8, p. 506] (boundary transition in sub integral expression. Chapter. Integrals, that
depend on the parameter), to our case the statement of Theorem 1 follows.
6. Estimate of approximation order in the case of two variables. For estimate of approxima-
tion order of operator F t, x(!), y(!)( ) by operator polynomial (14) we use the modules of continu-
ity, defined like that
!(", #) = !(#) = max
t ,z1,z2$[0,1]
sup
%x & %%x + %y & %%y '#
" t, z1, z2, %%x , %%y( ) & " t, z1, z2, %x , %y( ) ,
where ! is positive number.
Theorem 6. Let condition (18) hold and operator (13) is considered on the compact ! and
! " 0 , then
F t, x(!), y(!)( ) " Pn#[ ] F, x(!), y !( )( ) $ 2% 1+ #m
m + #m
&
'(
)
*+
.
Proof. So long as on the compact ! we have Ci, j m( ) vmi, j( ) x z1( ) , y z2( ) ,!( ) > 0 and
Pm
![ ] 1, x("), y "( )( ) = 1, so, one can write
F t, x(!), y(!)( ) " Pn#[ ] F, x(!), y !( )( ) ≤
332 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
≤ Ci, j (m)
j=0
m!i
" vmi, j( ) x(z1), y(z2 ),#( )
i=0
m
"
$2
%% f t, z1, z2, x(z1), y(z2 )( ) ! f t, z1, z2,
i
m
, j
m
&
'(
)
*+ dz1dz2 ≤
≤ Ci, j (m)
j=0
m!i
" vmi, j( ) x(z1), y(z2 ),#( )
i=0
m
"
$2
%% & f (t, z1, z2, ', '); x(z1) !
i
m
+
j
m
! y(z2 )
(
)*
+
,-
dz1dz2 ≤
≤ Ci, j (m)
j=0
m!i
" vmi, j( ) x(z1), y(z2 ),#( )
i=0
m
"
$2
%% ×
× 1+ 1
!
x(z1) "
i
m
+
j
m
" y(z2 )
#
$%
&
'(
#
$%
&
'(
)( f (t, z1, z2, *, *); !) dz1dz2 ≤
≤ Ci, j (m)
j=0
m!i
" vmi, j( ) x z1( ) , y z2( ) ,#( )
i=0
m
"
$2
%% 1+ 1
&
x(z1) !
i
m
+
j
m
! y(z2 )
'
()
*
+,
'
()
*
+,
dz2dz1-(&) . (20)
Here we have used the following properties of modules of continuity:
! ""x , ""y( ) # ! "x , "y( ) $ % ""x # "x + ""y # "y( ), ! "#( ) $ 1+ "( )! #( ) at ! > 0 .
Using the identities (16), from (20) we obtain
F t, x(!), y(!)( ) " Pn#[ ] F, x(!), y !( )( ) ≤
≤ !(") + 1
"
Cmi
i=0
m
#
0
1
$ vm(i)(x(z1)) x(z1) %
i
m
dz1!(") +
1
"
Cmj
j=0
m
#
0
1
$ vm( j )(y(z2 )) y(z2 ) %
j
m
dz1!(").
Then, we act in the same way as while proving Theorem 2. We have
F t, x(!), y(!)( ) " Pn#[ ] F, x(!), y !( )( ) $ 1+ 1
%
1+ #m
m + #m
&
'(
)
*+
,(%) .
It is logically to choose ! =
1+ "m
m + "m
, that leads to the completion of the theorem proof.
In case ! = 0 for Bernstein operator we will get
F t, x !( ) , y !( )( ) " Bm F, x !( ) , y !( )( ) # 2$ 1
m
%
&'
(
)* .
7. Asymptotic estimate of approximation error in case of two variables. Now we determine
asymptotic estimate for the error
Rm
![ ] F, x "( ) , y "( )( ) = F t, x "( ) , y "( )( ) # Pm![ ] F, x "( ) , y "( )( ).
The following theorem is valid.
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 333
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Theorem 7. Let Urysohn operator (13) be considered on the compact ! and satisfy condition
(18). For every fixed t, z1, z2 ![0,1] the second derivative
!2
!xi!y2"i
f t, z1, z2, x, y( ) , i = 0,1, 2 ,
continuous in each point x z1( ) , y z2( ){ } !" . Then, by ! = !(m)" 0 when m! " we have
an asymptotic formula for the error
Rm
![ ] F, x "( ) , y "( )( ) =
= Cmi Cm!i
j
j=0
m!i
" vmi, j( ) x(z1), y(z2 ),#( )
i=0
m
"
$2
%% f (t, z1, z2, x(z1), y(z2 )) ! f t, z1, z2,
i
m
, j
m
&
'(
)
*+
&
'(
)
*+
dz1dz2 =
= ! 1+ "m
1+ "
x(z1) 1! x(z1)( )
2m
#
$%&2
'' ((fxx t, z1, z2, x(z1), y(z2 )( ) + x(z1)y(z2 )
m
((fxy t, z1, z2, x(z1), y(z2 )( ) +
+
y(z2 ) 1! y(z2 )( )
2m
""fyy t, z1, z2, x(z1), y(z2 )( ) #
$%
dz1dz2 ! &m
'[ ] x(z1), y(z2 )( )
(2
)) dz1dz2 , (21)
where
lim
m!"
max #, #
m
, 1
m
$
%&
'
()
*1
+m
#[ ] x z( ) , y z( )( ) = 0 .
Proof. Due to [2] and taking into account (19) we’ll have
Rm
![ ] F, x("), y "( )( ) =
= Cmi Cm!i
j
j=0
m!i
" vmi, j( )(x(z1), y(z2 ),#)
i=0
m
"
$2
%% f (t, z1, z2, x(z1), y(z2 )) ! f t, z1, z2,
i
m
, j
m
&
'(
)
*+
&
'(
)
*+
dz1dz2 =
= ! 1+ "m
1+ "
x(z1) 1! x(z1)( )
2m
#
$%&2
'' ((fxx t, z1, z2, x(z1), y(z2 )( ) + x(z1)y(z2 )
m
((fxy t, z1, z2, x(z1), y(z2 )( ) +
+
y z2( ) 1! y z2( )( )
2m
""fyy t, z1, z2, x z1( ) , y z2( )( ) #
$
% dz1dz2 ! &m
'[ ] x z1( ) , y z2( )( )
(2
)) dz1dz2 ,
where
!m
"[ ] x z1( ) , y z2( )( ) = 1
2
Cmi Cm#i
j vmi, j( ) x z1( ) , y z2( ) ,"( )
j=0
m#i
$
i=0
m
$ ×
× x z1( ) ! i
m
"
#$
%
&'
2 (2
(x2
f t, z1, z2, x z1( ) + )
i
m
! x z1( )"
#$
%
&' , y(z2 ) + )
j
m
! y z2( )"
#$
%
&'
"
#$
%
&'
*
+
,
-
.
/
0/
–
334 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
–
!2
!x2
f t, z1, z2, x z1( ) , y z2( )( ) "
#
$ +
+ 2 i
m
! x(z1)
"
#$
%
&'
j
m
! y(z2 )
"
#$
%
&'
(2
(x(y
f t, z1, z2, x(z1) + )
i
m
! x(z1)
"
#$
%
&' , y(z2 ) + )
j
m
! y(z1)
"
#$
%
&'
"
#$
%
&'
*
+
, –
–
!2
!x!y
f t, z1, z2, x z1( ) , y z2( )( ) "
#
$ +
j
m
% y(z2 )
&
'(
)
*+
2
×
×
!2
!y2
f t, z1, z2, x z1( ) + "
i
m
# x z1( )$
%&
'
() , y z2( ) + "
j
m
# y z2( )$
%&
'
()
$
%&
'
()
*
+
, –
– !
2
!y2
f t, z1, z2, x z1( ) , y z2( )( ) "
#
$
%
&
'
('
, ! "(0,1) .
From this relation we obtain the estimate
!m
"[ ] x z1( ) , y z2( )( ) #
1
2
Cmi Cm$i
j vmi, j( ) x z1( ) , y z2( ) ,"( )
j=0
m$i
%
i=0
m
% ×
× x z1( ) ! i
m
"
#$
%
&'
2
(
)2
)x2
f t, z1, z2, *, *( ) ; x z1( ) ! i
m
+ y z2( ) ! j
m
"
#$
%
&'
+
,
-
-
.
/
0
0
1
2
3
43
+
+ 2 i
m
! x z1( ) j
m
! y z2( ) "
#2
#x#y
f t, z1, z2, $, $( ) ; x(z1) !
i
m
+ y(z2 ) !
j
m
%
&'
(
)*
+
,
-
-
.
/
0
0
+
+ j
m
! y z2( )"
#$
%
&'
2
(
)2
)y2
f (t, z1, z2, *, *); x(z1) !
i
m
+ y(z2 ) !
j
m
"
#$
%
&'
+
,
-
-
.
/
0
0
1
2
3
43
≤
≤ 1
2
Cmi Cm!i
j vmi, j( )
j=0
m!i
"
i=0
m
" x z1( ) , y z2( ) ,#( ) 1+ 1
$
x(z1) !
i
m
+ y(z2 ) !
j
m
%
&'
(
)*
+
,-
.
/0
×
× x(z1) !
i
m
+ y(z2 ) !
j
m
"
#$
%
&'
2
( D2 f ; )( ) ≤
≤ 2 Cmi Cm!i
j vmi, j( )
j=0
m!i
"
i=0
m
" x z1( ) , y z2( ) ,#( ) 1+ 2
$2
x(z1) !
i
m
2
+ y(z2 ) !
j
m
2%
&
'
(
)
*
+
,
-
-
.
/
0
0
1
2
3
43
5
6
3
73
1/2
×
× Cmi Cm!i
j vmi, j( )
j=0
m!i
"
i=0
m
" x z1( ) , y z2( ) ,#( ) x(z1) !
i
m
4
+ y(z2 ) !
j
m
4$
%
&
&
'
(
)
)
*
+
,
-,
.
/
,
0,
1/2
1 D2 f ; 2( ) ,
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 335
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
where
! D2 f ; "( ) = max
i=0,1,2
max
t ,z1,z2#[0,1]
!
$2
$xi$y2%i
f (t, z1, z2, x, y); "
&
'(
)
*+
.
Taking into account equalities (16) and (10) one can extend the previous inequality as follows:
!m
"[ ] x z1( ) , y z2( )( ) # 2 1
4
+ 2"$
%&
'
() 3"2 + 6"
m
$
%&
'
() +
1
m2
3
4
+ 7"$
%&
'
() +
1
m3
*
+,
-
./
1/2
×
× 1+ 1
!2
1+ "m
m + "m
#
$%
&
'(
1/2
) D2 f ; !( ) = 2 1
4
+ 2"*
+,
-
./ 3"2 + 6"
m
*
+,
-
./ +
1
m2
3
4
+ 7"*
+,
-
./ +
1
m3
#
$%
&
'(
1/2
×
× ! D2 f ; 1+ "m
m + "m
#
$%
&
'(
(22)
where the following value ! =
1+ "m
m + "m
is used. From here the validness of the theorem state-
ment follows.
Remark 3. In case ! = 0 the inequality (22) transforms into the following form:
!m
0[ ] x z1( ) , y z2( )( ) "
2
m
3
4
+
1
m
#
$%
&
'(
1/2
) D2 f ; 1
m
#
$%
&
'(
,
which corresponds to the Theorem 5. 1 from [2] at ! = 0 . But when ! > 0 , this theorem most
likely, same as in one variable case, is incorrect, though inequality (22) is just an upper estimate.
Let ! = "
1
m
. Whereas in this case polynomial Pm
! 1
m
"
#$
%
&' F, x((), y (( )( ) keeps the polynomial of
two variables of m-degree, then substituting Taylor series into (21) for f t, z1, z2,
i
m
, j
m
!
"#
$
%&
f t, z1, z2,
i
m
, j
m
!
"#
$
%& =
('1)p
p !
x(z1) '
i
m
!
"#
$
%&
(
(x
+ y(z2 ) '
j
m
!
"#
$
%&
(
(y
!
"#
$
%&
p
p=0
)
* f (t, z1, z2, x(z1), y(z2 )),
we obtain
Pm
! 1
m
"
#$
%
&' F, x((), y(()( ) = F t, x((), y(()( ) + (!1)n+1
(n + 1)!n=m
)
* Cmi Cm!i
j
j=0
m!i
* vmi, j( ) x(z1), y(z2 ), !
1
m
+
,-
.
/0i=0
m
*
12
22 ×
× x(z1) !
i
m
"
#$
%
&'
(
(x
+ y(z2 ) !
j
m
"
#$
%
&'
(
(y
"
#$
%
&'
n+1
f (t, z1, z2, x(z1), y(z2 )) dz2dz1 . (23)
336 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
It is impossible to make reasoning similar to one-dimensional case, because here the polynomi-
als vmi, j( ) x z1( ) , y z2( ) , ! 1
m
"
#$
%
&' , are not nonnegative. Therefore, let’s use the following designa-
tion:
ak,n+1!k
! 1
m
"
#$
%
&' x z1( ) , y z2( ) , ! 1
m
(
)*
+
,- =
= Cmi Cm!i
j vm(i, j )
j=0
m!i
"
i=0
m
" x(z1), y(z2 ), !
1
m
#
$%
&
'( x(z1) !
i
m
)
*+
,
-.
k
y(z2 ) !
j
m
)
*+
,
-.
n+1!k
to estimate of behavior of
!Rn
! 1
m
"
#$
%
&' F, x((), y (( )( ) = Pm
! 1
m
"
#$
%
&' F, x((), y(()( ) ! F t, x((), y(()( ) .
Then, formula (23) takes the form
Pm
! 1
m
"
#$
%
&' F, x((), y(()( ) = F t, x((), y(()( ) +
+ (!1)n+1
(n + 1)!n=m
"
# Cn+1k
k=0
n+1
# ak, n+1!k
! 1
m
$
%&
'
() x(z1), y(z2 ), !
1
m
*
+,
-
./
02
11 ×
×
!n+1
!xk!yn+1"k
f (t, z1, z2, x(z1), y(z2 )) dz2dz1. (24)
One can verify that following correlations occurs:
An+1
! 1
m
"
#$
%
&' = max
x,y(0
x+y)1
ak,n+1!k
! 1
m
"
#$
%
&' x, y, ! 1
m
*
+,
-
./ =
= max
0!x!1
ak,n+1"k
" 1
m
#
$%
&
'( x,1" x, " 1
m
)
*+
,
-. = max
0!x!1
an+1,0
" 1
m
#
$%
&
'( x,1" x, " 1
m
)
*+
,
-. , (25)
ak,n+1!k
! 1
m
"
#$
%
&' x,1! x, ! 1
m
(
)*
+
,- =
= mm!1
(m ! 1)!
x x !
1
m
"
#$
%
&'… x !
m
m
"
#$
%
&' (!1)iCmi x !
i
m
"
#$
%
&'
n+1
i=0
m
( , k = 0,1,…, n + 1.
Taking into account (25) from (24) we obtain the inequality
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 337
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Rn
! 1
m
"
#$
%
&' F, x((), y (( )( ) ≤
≤ 1
(n + 1)!n=m
!
" Cn+1k
k=0
n+1
" ak,n+1#k
# 1
m
$
%&
'
() x(z1), y(z2 ), #
1
m
*
+,
-
./
02
11 ×
× !n+1
!xk!yn+1"k
f (t, z1, z2, x(z1), y(z2 )) dz2dz1 #
2n+1
(n + 1)!
An+1
" 1
m
$
%&
'
() Dn+1 f
n=m
*
+ , (26)
where
Dn+1 f = max
0!t ,z1,z2 !1
z1+z2 !1
max
0!x(z1),y(z2 )!1
x(z1)+y(z2 )!1
"n+1
"xk"yn+1#k
f (t, z1, z2, x(z1), y(z2 )) .
Let’s consider the first (main) summand in a right-hand side of inequality (26). As,
ak,m+1!k
! 1
m
"
#$
%
&' x,1! x, ! 1
m
(
)*
+
,- =
mm!1
(m ! 1)!
x x !
1
m
(
)*
+
,-… x !
m
m
(
)*
+
,- (!1)iCmi x !
i
m
(
)*
+
,-
m+1
i=0
m
. =
= (m + 1) x !
1
2
"
#$
%
&' x x !
1
m
"
#$
%
&'… x !
m
m
"
#$
%
&'
then
Am+1
! 1
m
"
#$
%
&' = (m + 1) max
x([0,1]
x !
1
2
)
*+
,
-. x !
i
m
)
*+
,
-.i=0
m
/ =
=
m + 1
mm+2 max
t![0,m]
t " m
2
#
$%
&
'( t " i( )
i=0
m
) =
m + 1
mm+2 max
t![0,m]
*(t) . (27)
To estimate a right-hand side of relation (27) analogously to [11, p. 95], we’ll consider the func-
tion
!(t) = t " m
2
#
$%
&
'( (t " i)
i=0
m
) = ! z + m
2
#
$%
&
'( = z z2 " m
2
#
$%
&
'(
2*
+
,
,
-
.
/
/
z2 " m " 2
2
#
$%
&
'(
2*
+
,
,
-
.
/
/
…,
that as the function of z is even or odd with respect to evenness or oddness of m . The following
relation is valid:
!(t + 1) = v(t)!(t), v(t) = t + 1
t " m
t + 1" m
2
t " m
2
,
338 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
where function v(t) is negative, when t changes from 0 to m
2
! 1 and its module comes up to
maximum at point
tmax =
m ! 1! m + 1
2
,
that can be defined by the formula
max
0!t!m
2
"1
v(t) = v(tmax ) =
"1+ m + 1
1+ m + 1
#
$
%
&
'
(
2
. (28)
From (28) we obtain, that v(tmax ) is an increasing function relatively to m ! 3, 4,…{ } , its
maximum is reached in the infinity and is equal to 1. Thus, the extreme values of !(t) will de-
crease in module up to the middle of the interval [0,m], and then by symmetry (antisymmetry) will
increase. Above-stated give grounds to conclude: there exists the point x = ! " 0, 1
m
#
$%
&
'( , where the
equality holds
Am+1
! 1
m
"
#$
%
&' = (m + 1) ( !
1
2
)
*+
,
-. ( !
i
m
)
*+
,
-.i=0
m
/ .
Hence, the estimate follows
Am+1
! 1
m
"
#$
%
&' (
1
2
(m + 1)!
mm+1 .
Then we estimate Am+2
! 1
m
"
#$
%
&' . So, we have
Am+2
! 1
m
"
#$
%
&' =
mm
m !
max
0(x(1
x x !
1
m
)
*+
,
-.… x !
m
m
)
*+
,
-. (!1)iCmi x !
i
m
)
*+
,
-.
m+2
i=0
m
/ =
= 1
m !mm+3 max0!t!m
"(t) (#1)iCmi (t # i)m+2
i=0
m
$ = max
0!t!m
am+2,0
# 1
m
%
&'
(
)* t
m
,1# t
m
, # 1
m
+
,-
.
/0 ,
where !(t) = t(t " 1)(t " 2)…(t " m). The following relation is valid:
am+2,0
! 1
m
"
#$
%
&' t + 1
m
,1! t + 1
m
, ! 1
m
(
)*
+
,- = vm+2 (t)am+2,0
! 1
m
"
#$
%
&' t
m
,1! t
m
, ! 1
m
(
)*
+
,- ,
vm+2 (t) =
t + 1
t ! m
(!1)iCmi (t + 1! i)m+2
i=0
m"
(!1)iCmi (t ! i)m+2
i=0
m"
.
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 339
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
And the function vm+2 (t) meets the conditions
vm+2 (t) < 0, t ! 0, m " 1
2
#
$%
&
'(
, max
t! 0, m"1( )/2[ ]
vm+2 (t) ! = ! vm+2
m " 1
2
)
*+
,
-. ! = !1 ,
that can be evident by means of direct verification.
Rely on foresaid reasoning one can say that the point ! " 0, 1
m
#
$%
&
'( exists and the following cor-
relation is valid:
Am+2
! 1
m
"
#$
%
&' =
mm
m !
( ( !
1
m
)
*+
,
-.… ( !
m
m
)
*+
,
-. (!1)iCmi ( !
i
m
)
*+
,
-.
m+2
i=0
m
/ 0 2m .
Let’s show, that the same estimate will occur also for An+1
! 1
m
"
#$
%
&' ! n = m,!m + 1,!m + 2,!… . We
need to introduce the next function
fm,n (t) = (!1)iCmi (t ! i)n+1
i=0
m
" ,
that is followed by a valid correlation
d
dt
fm,n (t) = (n + 1) fm,n!1(t).
Then we use a certain integral representation of difference of arbitrary order (see, for instance
[11]), that has the consequence in a form
fm,n (t) = (!1)i
i=0
m
" Cmi (t ! i)n+1 =
m !(n + 1)!
(n ! m + 1)!
… (t ! z1 ! z2 !…! zm )n!m+1dzm…dz1
0
zm!1
#
0
z1
#
0
1
# . (29)
Let m + n be even number, then from (29) follows that the function fm,n (t) is increasing in
the interval [0,m]. But since in this case
fm,n
m
2
!
"#
$
%& = 0 , (30)
then fm,n (0) < fm,n (t) < 0 , t ! 0, m
2
"
#$
%
&' . This implies, that
0 <
fm,n (t + 1)
fm,n (t)
< 1, t ! 0, m
2
" 1#
$%
&
'(
. (31)
The validity of (30) follows from the equality
340 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
fm,n
m
2
!
"#
$
%& =
m
2
!
"#
$
%&
n+1
+ ('1)m '
m
2
!
"#
$
%&
n+1
+
+ Cm1 !
m
2
! 1"
#$
%
&'
n+1
+ (!1)m!1 !
m
2
+ 1"
#$
%
&'
n+1"
#
$
%
&
' +… = 0 .
Now let m + n be odd number, then the function fm,n (t) is a decreasing function in the inter-
val 0, m
2
!
"#
$
%& and the inequalities
0 < fm,n
m
2
!
"#
$
%& < fm,n (t) < fm,n (0), t ! 0, m
2
"
#$
%
&'
is valid. They also lead to statement (31), as in previous case. Established inequalities (31) lead to
the conclusion, that the function
vn+1(t) =
t + 1
t ! m
fm,n (t + 1)
fm,n (t)
=
t + 1
t ! m
(!1)iCmi (t + 1! i)n+1i=0
m"
(!1)iCmi (t ! i)n+1i=0
m"
is negative and module of it is less than 1 in the interval 0, m
2
! 1"
#$
%
&' . Then we have
An+1
! 1
m
"
#$
%
&' = (m + 1) max
x([0,1]
x !
i
m
)
*+
,
-.i=0
m
/ (!1)i
i=0
m
0 Cmi x !
i
m
)
*+
,
-.
n+1
=
=
m + 1
mm+2 max
t![0,m]
t " i( ) fm,n (t)
i=0
m
# =
m + 1
mm+2 max
t![0,m]
$(t) .
Hence, since !(t + 1) = t + 1
t " m
fm,n (t + 1)
fm,n (t)
!(t), then taking into account (31) one can state, that
maximum modulo values of function !(t) decreases in the interval 0, m
2
!
"#
$
%&
. So, there exists the
point like ! " 0, 1
m
#
$%
&
'( , that the following relation is valid:
An+1
! 1
m
"
#$
%
&' ! = !m
m
m !
( ( !
1
m
)
*+
,
-.… ( !
m
m
)
*+
,
-. (!1)iCmi ( !
i
m
)
*+
,
-.
n+1
i=0
m
/ ! 0 !2m . (32)
! n = m,!m + 1,!… .
The inequality (32) together with (26) lead to the estimate
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 341
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Rn
! 1
m
"
#$
%
&' F, x((), y (( )( ) ) 2m 2n+1
(n + 1)!
Dn+1 f
n=m
*
+ ,
which is followed by the following theorem.
Theorem 8. Let function f (t, z1, z2, x, y) be such, that in the domain 0 ! t ! 1, 0 ! z1, z2 ,
z1 + z2 ! 1, 0 ! x, y ! 1, x + y ! 1 there exist Dn+1 f , n = 1, 2,… such, the series
Dm+1 f
m + 1
+
2Dm+2 f
(m + 1)(m + 2)
+
22Dm+3 f
(m + 1)(m + 2)(m + 3)
+…
is convergent and its sum has upper estimate M , that doesn’t depend on m . Then, the following
inequalities are valid
Rn
! 1
m
"
#$
%
&' F, x((), y (( )( ) )
22m+1
m !
M )
2
*m
4e
m
+
,-
.
/0
m
M .
Example 2. Let’s consider Urysohn operator
F t, x !( ) , y !( )( ) = sin x z1( )( ) cos y z2( )( ) dz1dz2
"2
## , x z1( ) , y z2( ){ } !" ,
and construct operator polynomial (14) for it
Pm
![ ] F, x "( ) , y "( )( ) =
= m !
i ! j ! m ! i ! j( )! vm
i, j( ) x z1( ) , y z2( ) ,"( ) sin x z1( )( ) cos y z2( )( )
0#i+ j#m
$
%2
&& dz1dz2 ,
where
vmi, j( ) x z1( ) , y z2( ) ,!( ) =
=
x z1( ) + k1!( ) y z2( ) + k2!( ) 1" x z1( ) " y z2( ) + k3!( )k3=0
m"i" j"1#k2 =0
j"1#k1=0
i"1#
1+ k4!( )k4 =0
m"1#
, ! " 0 .
We choose, for example, ! = "
1
m
and x z1( ) = z1
1+ z12
, y z2( ) = z2
1+ z2
. For calculation
we’ll use Maple. The results we’ll write in Table 3, where
!1 = F t, x("), y "( )( ) # Pm1/m[ ] F, x("), y "( )( ) ,
!2 = F t, x("), y "( )( ) # Bm F, x("), y "( )( ) ,
!3 = F t, x("), y "( )( ) # Pm#1/m[ ] F, x("), y "( )( ) .
342 V. L. MAKAROV, I. I. DEMKIV
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Table 3. Results of calculations of Example 2.
n !1 C1 = m * !1 !2 C2 = m * !2 !3
1 0.013248971 0.013248972 0.1324897e-1 0.013248972 0.013248972
2 0.87424439e-2 0.017484888 0.6489178e-2 0.012978357 0.0002706
4 0.52128810e-2 0.020851524 0.3232767e-2 0.012931069 0.236968e-6
8 0.28857948e-2 0.023086359 0.1616030e-2 0.012928242 0.17266e-13
16 0.15249613e-2 0.024399381 0.8080102e-3 0.012928163 0.53346e-26
We see, that the inequalities
!1 "
0.03
m
, !2 "
0.015
m
, !3 "
22m
m !
(e2 # 1) M <
1
2
(e2 # 1)$
%&
'
()
are valid.
Example 3. Let’s consider Urysohn operator
F t, x !( ) , y !( )( ) = 1" 0.5z2x " 0.5z1y( )4 dz1dz2
#2
$$ , x z1( ) , y z2( ){ } !" .
We construct the operator polynomial (14) for it
Pm
![ ] F, x "( ) , y "( )( ) =
= m !
i ! j ! m ! i ! j( )! vm
i, j( ) x(z1), y(z2 ),"( ) 1! 0.5z2
i
m
! 0.5z1
1
m
#
$%
&
'(
4
0)i+ j)m
*
+2
,, dz1dz2 ,
where
vmi, j( ) x z1( ) , y z2( ) ,!( ) =
=
x z1( ) + k1!( ) y z2( ) + k2!( ) 1" x z1( ) " y z2( ) + k3!( )k3=0
m"i" j"1#k2 =0
j"1#k1=0
i"1#
1+ k4!( )k4 =0
m"1#
, ! " 0 .
Let’s choose, for example, ! =
1
m
, ! = 0 , ! = "
1
m
and x z1( ) = z1
1+ z12
, y z2( ) = z2
1+ z2
.
For calculation we use Maple. The results we write in Table 4, where
!1 = F t, x("), y "( )( ) # Pm1/m[ ] F, x("), y "( )( ) , !2 = F t, x("), y "( )( ) # Bm F, x("), y "( )( ) ,
!3 = F t, x("), y "( )( ) # Pm#1/m[ ] F, x("), y "( )( ) .
APPROXIMATION OF URYSOHN OPERATOR WITH OPERATOR POLYNOMIALS OF STANCU TYPE 343
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 3
Table 4. Results of calculations of Example 3.
n !1 C1 = m * !1 !2 C2 = m * !2 !3
1 0. 016387 0. 016387 0. 016387 0. 016387 0. 016387
2 0. 011145 0. 022290 0. 008524 0. 017048 0. 00066
4 0. 006819 0. 027274 0. 004347 0. 017386 0
8 0. 003848 0. 030780 0. 002195 0. 017557 0
16 0. 002058 0. 032934 0. 001103 0. 0176430 0
32 0. 001067 0. 034143 0. 000553 0. 017686 0
We see, that inequalities
!1 "
0.036
m
, !2 "
0.018
m
,
are valid and beginning from n = 4 the approximation error !3 is equal to zero.
1. Stancu D. D. Approximation of functions by anew class of linear polynomial operators // Rev. roum. math. pures et
appl. – 1968. – № 13. – P. 1173 – 1194.
2. Stancu D. D. A new class of uniform approximating polynomial operators in two and several variables // Proc. Conf.
Constructive Theory Functions (Approximation Theory) (Budapest, 1969). – Budapest: Akadémiai Kiadó. 1972. –
P. 443 – 455.
3. Demkiv I. I. On approximation of Urysohn operator with operator polynomials of Bernstein type // Visn. Lviv Univ.
Ser. Appl. Math. and Inform. – 2000. – № 2. – P. 26 – 30.
4. Makarov V. L., Khlobystov V. V. Interpolation method of identification problem solution for functional system which
is described with Urysohn operator // Dokl. AN USSR. – 1988. – 300, № 6. – P. 1332 – 1336.
5. Demkiv I. I. On approximation of Urysohn operator with operator polynomials of Bernstein type in the case two
variables // Visn. Lviv Polytech. Nat. Univ. Ser. Appl. Math. – 2000. – № 1. – P. 111 – 115.
6. Makarov V. L., Khlobystov V. V. On the identification of non-linear operators and its application // BEM IX. – 1987.
– № 1. – P. 43 – 58.
7. Baskakov V. Generalization of some theorems of P. Korovkin on positive operators // Math. Notes. – 1973. – 13, № 6.
– P. 785 – 794.
8. Fihtengoltz G. Coutse of differential and integral calculus. – М.: Fizmatlit, 2001. – Vol. 2. – 810 p.
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Received 18.11.11,
after revision — 01.03.12
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| id | umjimathkievua-article-2579 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:26:15Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a0/9f52b37e831a7c672a52f21e5765bfa0.pdf |
| spelling | umjimathkievua-article-25792020-03-18T19:30:02Z Approximation of Urysohn operator with operator polynomials of Stancu type Наближення оператора Урисона операторними поліномами типу станку Demkiv, I. I. Makarov, V. L. Демків, І. І. Макаров, В. Л. We study a one-parameter family of positive polynomial operators of one and two variables that approximate the Urysohn operator. In the case of two variables, the integration domain is a "rectangular isosceles triangle". As a special case, Bernstein-type polynomials are obtained. The Stancu asymptotic formulas for remainders are refined. Досліджується однопараметрична сім’я додатних поліноміальних операторів від однієї та двох змінних, що наближають оператор Урисона. У випадку двох змінних областю інтегрування є „прямокутний рівнобедрений трикутник”. Як окремий випадок, одержано поліноми типу Бернштейна. Дано уточнення асимптотичних формул Станку для залишкових членів. Institute of Mathematics, NAS of Ukraine 2012-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2579 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 3 (2012); 318-343 Український математичний журнал; Том 64 № 3 (2012); 318-343 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2579/1917 https://umj.imath.kiev.ua/index.php/umj/article/view/2579/1918 Copyright (c) 2012 Demkiv I. I.; Makarov V. L. |
| spellingShingle | Demkiv, I. I. Makarov, V. L. Демків, І. І. Макаров, В. Л. Approximation of Urysohn operator with operator polynomials of Stancu type |
| title | Approximation of Urysohn operator with operator polynomials of Stancu type |
| title_alt | Наближення оператора Урисона операторними поліномами типу станку |
| title_full | Approximation of Urysohn operator with operator polynomials of Stancu type |
| title_fullStr | Approximation of Urysohn operator with operator polynomials of Stancu type |
| title_full_unstemmed | Approximation of Urysohn operator with operator polynomials of Stancu type |
| title_short | Approximation of Urysohn operator with operator polynomials of Stancu type |
| title_sort | approximation of urysohn operator with operator polynomials of stancu type |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2579 |
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