Applied Mathematics E-Notes, 15(2015), 327-341 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Statistical Korovkin-Type Theory For
Matirx-Valued Functions Of Two Variables
Fadime Diriky, Kamil Demirciz
Received 3 June 2015
Abstract
In this work, we investigate an approximation problem for matrix valued positive linear operators of two variables. Also using the A-statistical convergence
which is stronger than Pringsheim convergence of double sequences, we prove a
Korovkin-type approximation theorem for matrix valued positive linear operators
of two variables. We also compute the rates of A-statistical convergence of this
operators.
1
Introduction
One of the most important and basic results in approximation theory is the classical
Bohman-Korovkin theorem (see for instance [10]). This theorem establishes the uniform convergence in the space C[a; b] of all the continuous real functions de…ned on
the interval [a; b], for a sequence of positive linear operators (Tn ) acting on C[a; b],
assuming the convergence only on the test functions 1; x; x2 . Of course, this theory
mainly gives the approximation a scaler-valued function by means of linear positive
operators. However, in [16], Serra-Capizzano presented a new Korovkin-type result
for matrix valued functions. Some other related topics may be found in the papers
[13, 17, 18] and cited therein. In particular, the use of statistical convergence had a
great impulse in recent years. Furthermore, with the help of the concept of uniform
statistical convergence, which is a regular (non-matrix) summability transformation,
various statistical approximation results have been proved [1, 2, 3, 5, 8, 9]. Then, it
was demonstrated that those results are more powerful than the classical Korovkin
theorem. In [4], using the statistical convergence, Duman and Erkuş-Duman proved
Korovkin-type approximation theorem for matrix valued positive linear operators. Our
primary interest in the present paper is to obtain a general Korovkin-type approximation theorem for double sequences of matrix valued positive linear operators of two
variables from C(D; Cs t ) to itself where D is a compact subset of R2 :
Mathematics Subject Classi…cations: 40G15, 41A36.
Author. Sinop University, Faculty of Arts and Sciences, Department of Mathematics, TR-57000, Sinop, Turkey.
z Sinop University, Faculty of Arts and Sciences, Department of Mathematics, TR-57000, Sinop,
Turkey.
y Corresponding
327
328
Matrix-Valued Functions of Two Variables
We begin with some de…nitions and notations which we will use in the sequel. As
usual, a double sequence
x = (xmn ) ; m; n 2 N;
is convergent in Pringsheim’s sense if, for every " > 0; there exists N = N (") 2 N such
that jxmn Lj < " whenever m; n > N . Then, L is called the Pringsheim limit of x
and is denoted by P limx = L (see [14]). In this case, we say that x = (xmn ) is
m;n
“P -convergent to L”. Also, if there exists a positive number M such that jxmn j M
for all (m; n) 2 N2 = N N; then x = (xmn ) is said to be bounded. Note that in
contrast to the case for single sequences, a convergent double sequence needs not to be
bounded.
Now let
A = [aprmn ]; p; r; m; n 2 N;
be a four-dimensional summability matrix. For a given double sequence x = (xmn ),
the A-transform of x, denoted by Ax := f(Ax)pr g, is given by
(Ax)pr =
X
aprmn xmn ;
(m;n)2N2
p; r 2 N;
provided the double series converges in Pringsheim’s sense for every (p; r) 2 N2 . In
summability theory, a two-dimensional matrix transformation is said to be regular if it
maps every convergent sequence in to a convergent sequence with the same limit. The
well-known characterization for two dimensional matrix transformations is known as
Silverman-Toeplitz conditions (see, for instance, [7]). In 1926, Robinson [15] presented
a four dimensional analog of the regularity by considering an additional assumption
of boundedness. This assumption was made because a double P -convergent sequence
is not necessarily bounded. The de…nition and the characterization of regularity for
four dimensional matrices is known as Robison-Hamilton conditions, or brie‡y, RHregularity (see, [6, 15]).
Recall that a four dimensional matrix A = [aprmn ] is said to be RH-regular if it
maps every bounded P -convergent sequence into a P -convergent sequence with the
same P -limit. The Robison-Hamilton conditions state that a four dimensional matrix
A = [aprmn ] is RH-regular if and only if
(i) P
lim aprmn = 0 for each (m; n) 2 N2 ;
(ii) P
lim
(iii) P
lim
(iv) P
lim
(v)
p;r
P
p;r (m;n)2N2
P
p;r m2N
P
p;r n2N
P
(m;n)2N2
aprmn = 1,
japrmn j = 0 for each n 2 N,
japrmn j = 0 for each m 2 N,
japrmn j is P convergent,
F. Dirik and K. Demirci
329
(vi) there exist …nite positive integers A and B such that
P
m;n>B
for every (p; r) 2 N2 :
K
japrmn j < A holds
Now let A = [aprmn ] be a non-negative RH-regular summability matrix, and let
N2 . Then A-density of K is given by
X
(2)
lim
aprmn
A fKg := P
p;r
(m;n)2K
provided that the limit on the right-hand side exists in Pringsheim’s sense. A real
double sequence x = (xmn ) is said to be A statistically convergent to a number L if,
for every " > 0,
(2)
2
Lj "g = 0.
A f(m; n) 2 N : jxmn
(2)
In this case, we write stA
limxmn = L. We should note that if we take A = C(1; 1),
m;n
which is the double Cesáro matrix, then C(1; 1)-statistical convergence coincides with
the notion of statistical convergence for double sequence, which was introduced in
[11, 12]. Finally, if we replace the matrix A by the identity matrix for four-dimensional
matrices, then A-statistical convergence reduces to the Pringsheim convergence.
A P -convergent double sequence is A-statistically convergent to the same value but
the converse does not hold true.
2
A Korovkin-Type Approximation Theorem for Double Sequences
In this section, we give a Korovkin-type theorem for double sequences of matrix valued positive linear operators de…ned on C(D; Cs t ) using the concept of A-statistical
convergence.
Let s; t be two …xed natural numbers and D a compact subset of R2 . By C(D; Cs t )
we denote the space of all continuous functions F acting on D and having values in
the space Cs t of the complex s t matrices such that
F (x; y) := [fjk (x; y)]s
t
; ((x; y) 2 D; 1
j
s; 1
k
t),
(1)
where the symbol [bjk ]s t denotes the s t matrix. Here, by the continuity of F we
mean that all scalar valued functions fjk are continuous on D. Then, the norm k ks t
on the space C(D; Cs t ) is de…ned by
!
kF ks
t
:=
max
1 j s; 1 k t
kfjk k :=
max
1 j s; 1 k t
sup jfjk (x; y)j :
(x;y)2D
Throughout this paper we use the following test functions
E0jk (u; v) = Ejk ; E1jk (u; v) = uEjk ;
(2)
330
Matrix-Valued Functions of Two Variables
E2jk (u; v) = vEjk and E3jk (u; v) = u2 + v 2 Ejk ;
(3)
for (u; v) 2 D, 1 j
s, 1 k
t; where Ejk denotes the matrix of the canonical
basis of Cs t being 1 in the position (j; k) and zero otherwise.
Let : C(D; Cs t ) ! C(D; Cs t ) be an operator, and let us assume that
(i)
(ii)
(aF + bG) = a (F ) + b (G) for any ;
2 C and F; G 2 C(D; Cs t ),
(F ) K (jF j) for any function F 2 C(D; Cs t ) and for a …xed positive constant K.
Under the above-mentioned assumptions, the operator is said to be a matrix linear
positive operator, or brie‡y, mLP O (see, for details, [16]). The inequality appearing
in (ii) is understood to be componentwise, i.e., holding for any component (j; k) 2
f1; 2; :::; sg f1; 2; :::; tg:
Now we have the following main result.
THEOREM 1. Let A = [aprmn ] be a nonnegative RH-regular summability matrix
and let ( mn ) be a double sequence of mP LOs acting from C(D; Cs t ) into itself.
Then for all (j; k) 2 f1; 2; :::; sg f1; 2; :::; tg and for each i = 0; 1; 2; 3,
st2A
lim k
m;n
mn (Eijk )
Eijk ks
t
= 0,
(4)
where Eijk is given by (2) and (3), if and only if for every F 2 C(D; Cs t ) as in (1);
st2A
lim k
m;n
mn (F )
F ks
t
= 0.
(5)
PROOF. Since each Eijk 2 C(D; Cs t ), (i = 0; 1; 2; 3), the implication (5))(4) is
obvious. Suppose now that (4) holds. Let F 2 C(D; Cs t ) and (x; y) 2 D be …xed.
Fist, we calculate the expression j nm (F ; x; y) F (x; y)j, the symbol jBj denotes the
matrix having entries equal to the absolute value of the entries of the matrix B. We
can show that F (x; y) := [fjk (x; y)]s t ; 1 j s; 1 k t, can be written as follows:
F (x; y) :=
t
s X
X
fjk (x; y)Ejk =
j=1 k=1
s X
t
X
fjk (x; y)E0jk (u; v):
j=1 k=1
Hence, we obtain
mn (F (x; y); x; y) =
s X
t
X
fjk (x; y)
mn
(E0jk (u; v); x; y) :
(6)
j=1 k=1
Also, the continuity of fjk on D; for a given " > 0, there exists a number
that for all (u; v) 2 D satisfying
q
2
2
(u x) + (v y)
:
> 0 such
F. Dirik and K. Demirci
331
We have
jfjk (u; v)
fjk (x; y)j
" for 1
Also we get for all (x; y) ; (u; v) 2 D satisfying
jfjk (u; v)
where M :=
2Mjk
fjk (x; y)j
2
j
s; 1
q
(u
t:
2
j
y) >
s; 1
2
(7)
2
x) + (v
'(u; v) for 1
sup jfjk (x; y)j and '(u; v) = (u
k
k
that
t;
(8)
2
x) + (v
y) . Combining (7) and
(x;y)2D
(8) we have for (u; v) 2 D that
jfjk (u; v)
fjk (x; y)j
"+
2Mjk
'(u; v):
(9)
'(u; v)E;
(10)
2
Then observe that, by (9),
jF (u; v)
where E is the s
mn
"E +
2M
2
t matrix such that all entires 1, and
M :=
Since
F (x; y)j
max
Mjk = kF ks
1 j s; 1 k t
t
:
is a mP LO, from (6) and (10), we get
j
K
mn (F (u; v); x; y)
F (x; y)j ; x; y)
mn (jF (u; v)
s X
t
X
("K + M )
j=1 k=1
+
F (x; y)j
2KM
2
j
mn
+j
mn (F (x; y); x; y)
(E0jk (u; v); x; y)
mn ('(u; v)E; x; y)
E0jk (x; y)j
+ "KE
(11)
for a …xed positive constant K. Now we compute the expression
on the left-hand side of (11). By a simple calculation, we get
mn ('(u; v)E; x; y)
=
F (x; y)j
mn ('(u; v)E; x; y)
s X
t
X
(
'(u; v)Ejk ; x; y)
mn
j=1 k=1
=
s X
t
X
j=1 k=1
2y
f
mn
mn
(E3jk ; x; y)
2x
mn
(E2jk ; x; y) + x2 + y 2
(E1jk ; x; y)
mn
(E0jk ; x; y) :
332
Matrix-Valued Functions of Two Variables
Then,
mn (
s X
t
X
j=1 k=1
+2A
2
(u
j
s X
t
X
s X
t
X
j=1 k=1
y)
E; x; y)
(E3jk ; x; y)
mn
j=1 k=1
+2B
2
x) + (v
E3jk (x; y)j
j
mn
(E1jk ; x; y)
E1jk (x; y)j
j
mn
(E2jk ; x; y)
E2jk (x; y)j
s X
t
X
+ A2 + B 2
j=1 k=1
j
mn
(E0jk ; x; y)
E0jk (x; y); j
(12)
where A := max jxj and B := max jyj. Combining (11) and (12), we obtain
j
mn (F (u; v); x; y)
s X
t
X
("K + M )
j=1 k=1
+"KE +
j
F (x; y)j
mn
(E0jk (u; v); x; y)
t
s
4AKM X X
2
j=1 k=1
+
t
s
4BKM X X
2
j=1 k=1
+
s
t
2KM X X
2
j=1 k=1
+
j
j
mn
mn
j
mn
(E3jk ; x; y)
2
j=1 k=1
"KE + C
t X
3
s X
X
j=1 k=1 i=0
where
(E1jk ; x; y)
(E2jk ; x; y)
t
s
2 A2 + B 2 KM X X
j
mn
j
E0jk (x; y)j
mn
E1jk (x; y)j
E2jk (x; y)j
E3jk (x; y)j
(E0jk (u; v); x; y)
(Eijk ; x; y)
(
E0jk (x; y)j
Eijk (x; y)j ;
2 A2 + B 2 KM 4AKM 4BKM 2KM
C := max "K + M +
;
;
;
2
2
2
2
)
:
Then, taking maximum of all entries of the corresponding matrices and taking supremum over (x; y) 2 D; we get
8
9
s X
t X
3
<X
=
k mn (F ) F ks t "K + C
k mn (Eijk ) Eijk ks t :
(13)
:
;
j=1 k=1 i=0
F. Dirik and K. Demirci
333
0
0
"
K.
Now, for a given " > 0, choose " > 0 such that " <
n
:= (m; n) 2 N2 : k
and
ijk
:=
(
(m; n) 2 N2 : k
mn
mn (F )
(Eijk )
Then, de…ne
F ks
t
"
"
Eijk ks
t
0
o
0
"K
4stC
)
,
where i = 0; 1; 2; 3 and 1
j
s; 1
k
t: Then it is easy to see that
t S
3
s S
S
2
ijk . Thus, we may write, for every (p; r) 2 N , that
j=1 k=1 i=0
X
s X
t X
3
X
aprmn
X
j=1 k=1 i=0 (m;n)2
(m;n)2
aprmn :
ijk
Letting p; r ! 1, using (4), we obtain (5). The proof is complete.
3
Concluding Remarks
In Theorem 1 if we replace the matrix A by the double Cesáro matrix C(1; 1), then we
immediately get the following statistical result.
COROLLARY 1. Let ( mn ) be a double sequence of mP LOs acting from C(D; Cs t )
into itself. Then for all (j; k) 2 f1; 2; :::; sg f1; 2; :::; tg and for each i = 0; 1; 2; 3,
st2
lim k
m;n
mn (Eijk )
Eijk ks
t
= 0,
where Eijk is given by (2) and (3), if and only if for every F 2 C(D; Cs t ) as in (1);
st2
lim k
m;n
mn (F )
F ks
t
= 0.
In Theorem 1 if we replace the matrix A by the double identity matrix I, then we
immediately get the following classical result.
COROLLARY 2. Let ( mn ) be a double sequence of mP LOs acting from C(D; Cs t )
into itself. Then for all (j; k) 2 f1; 2; :::; sg f1; 2; :::; tg and for each i = 0; 1; 2; 3,
P
lim k
m;n
mn (Eijk )
Eijk ks
t
= 0,
where Eijk is given by (2) and (3), if and only if for every F 2 C(D; Cs t ) as in (1);
P
lim k
m;n
mn (F )
F ks
t
= 0.
334
Matrix-Valued Functions of Two Variables
Now we present an example such that our new approximation result works but
its classical case (Corollary 2) does not work. Let a, b; c; d be …xed real numbers
and D = [a; b] [c; d]. First consider the following the matrix-valued Bernstein-type
operators:
Bmn (F ; x; y)
m X
n
X
=
F
l
(b
m
a+
l=0 r=0
x
b
l
a
a
y
d
a) ; c +
r
c
c
b
b
r
(d
n
m l
x
a
m
l
c)
d
d
y
c
n
r
n r
;
(14)
where (x; y) 2 D, F 2 C (D; Cs t ) such that F (x; y) := [fjk (x; y)]s t ; 1
j
s;
1 k t. Also, observe that the matrix-valued Bernstein-type polynomials Bmn can
be also written as follows:
Bmn (F ; x; y)
m X
n X
s X
t
X
=
fjk a +
l=0 r=0 j=1 k=1
x
b
a
a
l
y
d
l
(b
m
r
c
c
b
b
r
(d
n
a) ; c +
m l
x
a
d
d
y
c
m
l
c)
n
r
n r
Ejk ;
(15)
where Ejk denotes the matrix of the canonical basis of Cs t being 1 in the position (j; k)
and zero otherwise. Then, by (14) or (15), we obtain that, for (j; k) 2 f1; 2; :::; sg
f1; 2; :::; tg;
Bmn (E0jk ; x; y)
m X
n
X
m n
=
l
l=0 r=0
x
b
r
l
(b
m
E0jk a +
m X
n
X
m
l
r=0
=
Ejk
=
E0jk (x; y) :
a) ; c +
n
r
l=0
l
a
a
x
b
y
d
r
(d
n
a
a
=
l=0 r=0
l
l
y
d
a
a
l
y
d
l
r
(b a) ; c + (d
m
n
= E1jk (x; y)
E1jk a +
= xEjk
r
x
b
b
b
x
a
m l
b
b
x
a
d
d
y
c
n r
m l
d
d
y
c
c)
Similarly, we get that, for (j; k) 2 f1; 2; :::; sg
Bmn (E1jk ; x; y)
m X
n
X
m n
r
c
c
c
c
r
n r
f1; 2; :::; tg;
c
c
r
b
b
x
a
m l
c)
and
Bmn (E2jk ; x; y) = yEjk = E2jk (x; y) :
d
d
y
c
n r
F. Dirik and K. Demirci
335
Finally, we obtain that, for (j; k) 2 f1; 2; :::; sg
=
Bmn (E3jk ; x; y)
m X
n
X
m n
l=0 r=0
l
x
b
r
a
a
l
y
d
f1; 2; :::; tg;
r
c
c
b
b
x
a
m l
r
(d c)
n
1
(b a) (x a)
2
=
x2
(x a) +
m
m
1
(d
c)
(y c)
2
+y 2
(y c) +
Ejk
n
n
1
1
(d
(b a) (x a)
2
2
(x a)
(y c) +
=
m
m
n
+E3jk (x; y) :
E3jk a +
l
(b
m
d
d
y
c
n r
a) ; c +
c) (y
n
c)
Ejk
Now take A = C(1; 1) and de…ne a double sequence u = (umn ) by
umn :=
1
0
if m and n are squares,
otherwise.
Using polynomials given by (14) and the double sequence u = (umn ), we introduce the
following mP LOs on C (D; Cs t ) :
mn (F ; x; y)
= (1 + umn )Bmn (F ; x; y) ;
(16)
where (m; n) 2 N2 , (x; y) 2 D and F 2 C (D; Cs t ) such that F (x; y) := [fjk (x; y)]s t ;
1 j
s; 1 k
t. So, using the properties of matrix-valued the above operators
given by (14), for each 1 j
s; 1 k
t, one can obtain the following results at
once:
k mn (Eijk ) Eijk ks t = umn , i = 0; 1; 2;
and
k
mn (E3jk )
E3jk ks
t
1
(
m
(d
+
2
a) +
c) (
n
umn
(
m
umn
+
(
n
+
where
= max jxj,
st2
= max jyj. Since st
lim k
m;n
mn (Eijk )
(b
a) (
m
a)
+
1
(
n
c)
2
(b
2
(d
a) +
c) +
a) (
m
c) (
n
a)
c)
umn
umn ;
limumn = 0, we conclude that
m;n
Eijk ks
t
= 0; i = 0; 1; 2; 3;
c)
2
336
Matrix-Valued Functions of Two Variables
for each 1
j
s; 1
k
t: So, by Theorem 1, we immediately see that
st2
lim k
mn (F )
m;n
F ks
t
=0
for all F 2 C (D; Cs t ) : However, since u is not ordinary convergent to zero, the double
sequence ( mn ) given by (16) does not satisfy the conditions of Corollary 2.
4
Rate of Convergence
Various ways of de…ning rates of convergence in the A-statistical sense for four-dimensional
summability matrices were introduced in [2]. In this section, we compute the corresponding rates of A-statistical convergence in Theorem 1 by means of two di¤erent
ways.
DEFINITION 1 ([2]). Let A = [aprmn ] be a non-negative RH-regular summability
matrix and let ( mn ) be a positive non-increasing double sequence. A double sequence
x = (xmn ) is A-statistically convergent to a number L with the rate of o( mn ) if for
every " > 0,
X
1
aprmn = 0,
P lim
p;r
pr
(m;n)2K(")
where
K(") := (m; n) 2 N2 : jxmn
Lj
" :
In this case, we write
xmn
(2)
L = stA
o(
mn )
as m; n ! 1:
DEFINITION 2 ([2]). Let A = [aprmn ] and ( mn ) be the same as in De…nition 1.
Then, a double sequence x = fxmn g is A-statistically convergent to a number L with
the rate of omn ( mn ) if for every " > 0,
X
P lim
aprmn = 0,
p;r
(m;n)2M (")
where
M (") := (m; n) 2 N2 : jxmn
Lj
"
mn
:
In this case, we write
xmn
(2)
L = stA
omn (
mn )
as m; n ! 1:
We see from the above statements that, in De…nition 1 the rate sequence ( mn )
directly e¤ects the entries of the matrix A = [aprmn ] although, according to De…nition
2, the rate is more controlled by the terms of the sequence x = (xmn ) (see for details,
[5])
F. Dirik and K. Demirci
337
Let F 2 C(D; Cs t ) such that
F (x; y) := [fjk (x; y)]s
; 1
t
j
s; 1
k
t.
Consider the the following modulus of continuity !(fjk ; ):
! (fjk ; ) := sup jfjk (u; v)
fjk (x; y)j : (u; v) ; (x; y) 2 D;
q
where fjk are scalar valued functions continuous on D and
matrix modulus of continuity of F as follows:
!s
(F ; ) :=
t
max
(u
2
x) + (v
y)
2
> 0: Then, we de…ne the
! (fjk ; ) :
1 j s; 1 k t
In order to obtain our result, we will make use of the elementary inequality, for all
F 2 C(D; Cs t ) and for ; > 0,
!s
t
(F ;
)
(1 + [ ]) ! s
t
(F ; ) ;
(17)
where [ ] is de…ned to be the greatest integer less than or equal to .
Then we have the following result.
THEOREM 2. Let A = [aprmn ] be a nonnegative RH-regular summability matrix, let ( mn ) be a positive non-increasing double sequence. and let ( mn ) be a
double sequence of mP LOs acting from C(D; Cs t ) into itself. Then for all (j; k) 2
f1; 2; :::; sg f1; 2; :::; tg and for each i = 0; 1; 2; 3,
(a) k
(b) ! s
mn (E0jk )
t
(F ;
E0jk ks
mn )
(2)
= stA
(2)
t
o(
= stA
o(
mn )
jk (u; v)
= (u
2
as m; n ! 1,
as m; n ! 1 where
F 2 C(D; Cs t ) and
where
mnjk )
x) + (v
mn
v
u s t
uX X
:= t
k
mn (
jk )ks t
j=1 k=1
2
y) for each (x; y) ; (u; v) 2 D:
Then, we get, for each F 2 C(D; Cs t ) as in (1);
k
mn (F )
F ks
(2)
t
= stA
o(
mn );
where
m;n
:=
max
1 j s; 1 k t
f
mnjk ; mn g
for all (m; n) 2 N2 . Furthermore, similar conclusions hold with the symbol “o”replaced
by “omn ”
338
Matrix-Valued Functions of Two Variables
PROOF. To see this, we …rst assume that (x; y) 2 D and F 2 C(D; Cs t ) be …xed,
and that (a) and (b) hold. Since mn is a mP LO, we get
j
mn (F (u; v); x; y)
K
mn (jF (u; v)
F (x; y)j
F (x; y)j ; x; y) + j
mn (F (x; y); x; y)
F (x; y)j ;
where K is a positive constant. Also,
jF (u; v)
F (x; y)j
!s
t
1+
where E is the s
write
j
F;
q
(u
2
2
(u
2
x) + (v
2
x) + (v
y)
2
y)
!
!s
E
t
(F ; ) E;
t matrix such that all entires 1. As in the proof Theorem 1, we may
mn (F (x; y); x; y)
F (x; y)j
t
s X
X
M
j=1 k=1
j
mn
(E0jk (u; v); x; y)
E0jk (x; y)j ; (19)
where
M :=
max
Mjk = kF ks
1 j s; 1 k t
t
:
By (18) and (19), we obtain
j
mn (F (u; v); x; y)
K! s
t
(F ; )
mn
F (x; y)j
(E) +
K
2 !s
t
(F ; )
s X
t
X
mn (
jk ; x; y)
j=1 k=1
+M
t
s X
X
j=1 k=1
K! s
t
j
(F ; )
mn
(E0jk (u; v); x; y)
t
s X
X
j=1 k=1
+M
t
s X
X
j=1 k=1
+
(18)
K
2 !s t
j
mn
(F ; )
j
mn
(E0jk (u; v); x; y)
(E0jk (u; v); x; y)
s X
t
X
E0jk (x; y)j
mn (
E0jk (x; y)j
E0jk (x; y)j
jk ; x; y)
+ K! s
t
(F ; ) E:
j=1 k=1
Taking supremum over (x; y) 2 D on the both-sides of the above inequality and
v
u s t
uX X
:= mn := t
k mn ( jk )ks t ;
j=1 k=1
F. Dirik and K. Demirci
339
then we obtain
k
mn (F )
F ks
2K! s
t
t
+K! s
(F ;
t
mn )
(F ;
mn )
s X
t
X
j=1 k=1
+M
s X
t
X
j=1 k=1
k
mn
k
mn
(E0jk ; x; y)
(E0jk ; x; y)
E0jk ks
t
E0jk ks
t
:
Hence, we get
k
B
+
mn (F )
8
<
:
!s
F ks
t (F ;
t
mn ) + ! s
t (F ;
mn )
s X
t
X
j=1 k=1
s X
t
X
j=1 k=1
k
mn
(E0jk ; x; y)
k
mn
(E0jk ; x; y)
E0jk ks
t
9
=
E0jk ks
(20)
t;
where B = max f2K; M g. Now, given " > 0, de…ne the following sets:
:= (m; n) 2 N2 : k
1
jk
:=
(m; n) 2 N2 : ! s
(m; n) 2 N2 : ! s
jk
where 1
:=
:=
j
t
(F ;
(m; n) 2 N2 : k
s; 1
k
mn ) k
mn
mn (F )
t
(F ;
mn
mn )
(E0jk ; x; y)
U :=
" ;
t
"
(2st + 1) B
(E0jk ; x; y)
E0jk ks
E0jk ks
t
;
t
"
(2st + 1) B
"
(2st + 1) B
,
t. Then, it follows from (20) that
1 0
1
0
t
s [
t
s [
[
[
@
A[@
A:
1[
jk
jk
j=1 k=1
Also, de…ning
F ks
(m; n) 2 N2 : ! s
j=1 k=1
t (F ;
mn )
r
"
(2st + 1) B
and
Ujk :=
2
(m; n) 2 N : k
mn
(E0jk ; x; y)
E0jk ks
t
r
"
(2st + 1) B
;
;
340
we have
Matrix-Valued Functions of Two Variables
jk
U [ Ujk ; which yields
0
1 0
s [
t
s [
t
[
[
@
Ujk A [ @
1[U [
j=1 k=1
Therefore, since
:=
mn
max
1 j s; 1 k t
N2 ,
X
1
f
pr
+
X
(m;n)2
1
pr
j=1 k=1
mnjk ; mn g ;
A:
we conclude that, for all (p; r) 2
aprmn
pr (m;n)2
1
jk
1
aprmn +
j=1 k=1
1
X
s X
t
X
aprmn +
0
@
s X
t
X
j=1 k=1
(m;n)2U
0
@
1
mnjk
X
(m;n)2Ujk
1
mnjk
X
(m;n)2
1
aprmn A
jk
1
aprmn A :
(21)
Letting p; r ! 1 (in any manner) on both sides of (21), from (18) and (19), we get
P
lim
p;r
1
X
aprmn = 0:
pr (m;n)2
Therefore, the proof is completed.
References
[1] K. Demirci and F. Dirik, A Korovkin-type approximation theorem for functions
of two variables in statistical sense, Turk J Math., 47(2010), 73–83.
[2] K. Demirci and F. Dirik, Four-dimensional matrix transformation and rate of Astatistical convergence of periodic functions, Math. Comput. Modelling, 52(2010),
1858–1866.
[3] K. Demirci and F. Dirik, Equi-ideal convergence of positive linear operators for
analytic p-ideals, Math. Commun., 16(2011), 169–178.
[4] O. Duman and E. Erkuş-Duman, Statistical Korovkin-type theory for matrixvalued functions, Stud. Sci. Math. Hung., 48(2011), 489–508.
[5] O. Duman, M. K. Khan and C. Orhan, A-Statistical convergence of approximating
operators, Math. Inequal. Appl., 4(2003), 689–699.
[6] H. J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2(1936),
29–60.
[7] G. H. Hardy, Divergent Series, Oxford Univ. Press, London, 1949.
F. Dirik and K. Demirci
341
[8] S. Karakuş and K. Demirci, Summation process of Korovkin type approximation
theorem, Miskolc Math. Notes, 12(2011), 75–85.
[9] S. Karakuş, K. Demirci and O. Duman, Equi-statistical convergence of positive
linear operators, J. Math. Anal. Appl., 339(2008), 1065–1072.
[10] P. P. Korovkin, Linear Operators and Approximation Theory, Translated from the
Russian ed. (1959). Russian Monographs and Texts on Advanced Mathematics
and Physics, Vol. III. Gordon and Breach Publishers, Inc., New York; Hindustan
Publishing Corp. (India), Delhi 1960.
[11] F. Moricz, Statistical convergence of multiple sequences, Arch. Math. (Basel),
81(2004), 82–89.
[12] Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J.
Math. Anal. Appl., 288(2003), 223–231.
[13] D. Noutsos, S. Serra-Capizzano and P. Vassalos, Two-level Toeplitz preconditioning: approximation results for matrices and functions, SIAM J. Sci. Comput.,
28(2006), 439–458.
[14] A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann.,
53(1900), 289–321.
[15] G. M. Robison, Divergent double sequences and series, Amer. Math. Soc. Transl.,
28(1926), 50–73.
[16] S. S. Capizzano, A Korovkin based approximation of multilevel Toeplitz matrices
(with rectangular unstructured blocks) via multilevel trigonometric matrix spaces,
SIAM J. Numer. Anal., 36(1999), 1831–1857.
[17] S. S. Capizzano, A Korovkin-type theory for …nite Toeplitz operators via matrix
algebras, Numer. Math., 82(1999), 117–142.
[18] S. S. Capizzano, Korovkin tests, approximation, and ergodic theory, Math. Comp.,
69(2000), 1533–1558.