MATEMATIQKI VESNIK
originalni nauqni rad
research paper
67, 4 (2015), 288–300
December 2015
KOROVKIN TYPE APPROXIMATION THEOREM
IN AI2 -STATISTICAL SENSE
Sudipta Dutta and Pratulananda Das
Abstract. In this paper we consider the notion of AI
2 -statistical convergence for real double
sequences which is an extension of the notion of AI -statistical convergence for real single sequences
introduced by Savas, Das and Dutta. We primarily apply this new notion to prove a Korovkin
type approximation theorem. In the last section, we study the rate of AI
2 -statistical convergence.
1. Introduction and background
Throughout the paper N will denote the set of all positive integers. Approximation theory has important applications in the theory of polynomial approximation
in various areas of functional analysis. For a sequence {Tn }n∈N of positive linear
operators on C(X), the space of real valued continuous functions on a compact
subset X of real numbers, Korovkin [20] first established the necessary and sufficient conditions for the uniform convergence of {Tn f }n∈N to a function f by using
the test functions e0 = 1, e1 = x, e2 = x2 [1]. The study of the Korovkin type
approximation theory has a long history and is a well-established area of research.
As is mentioned in [12] in particular, the matrix summability methods of Cesáro
type are strong enough to correct the lack of convergence of various sequences of
positive linear operators such as the interpolation operators of Hermite-Fejér [6]. In
recent years, using the concept of uniform statistical convergence various statistical
approximation results have been proved [9,10]. Erkuş and Duman [15] studied a
Korovkin type approximation theorem via A-statistical convergence in the space
Hw (I 2 ) where I 2 = [0, ∞) × [0, ∞) which was extended for double sequences of positive linear operators of two variables in A-statistical sense by Demirci and Dirik
in [12]. Our primary interest in this paper is to obtain a general Korovkin type
approximation theorem for double sequences of positive linear operators of two
variables from Hw (K) to C(K) where K = [0, A] × [0, B], A, B ∈ (0, 1), in the sense
of AI2 -statistical convergence.
2010 Mathematics Subject Classification: 40A35, 47B38, 41A25, 41A36
Keywords and phrases: Ideal; AI
2 -statistical convergence; positive linear operator; Korovkin
type approximation theorem; rate of convergence.
288
Korovkin type approximation
289
The concept of convergence of a sequence of real numbers was extended to
statistical convergence by Fast [17]. Further investigations started in this area after
the pioneering works of Šalát [31] and Fridy [18]. The notion of I-convergence of
real sequences was introduced by Kostyrko et al. [23] as a generalization of statistical
convergence using the notion of ideals (see [3,4,5] for further references). Later the
idea of I-convergence was also studied in topological spaces in [24]. On the other
hand statistical convergence was generalized to A-statistical convergence by Kolk
[21,22]. Later a lot of works have been done on matrix summability and A-statistical
convergence (see [2,7,8,11,16,19,21,22,25,29]). In particular, very recently in [33]
and [34] the two above mentioned approaches were unified and the very general
notion of AI -statistical convergence was introduced and studied. In this paper
we consider an extension of this notion to double sequences, namely AI2 -statistical
convergence.
A real double sequence {xmn }m,n∈N is said to be convergent to L in Pringsheim’s sense if for every ε > 0 there exists N (ε) ∈ N such that |xmn − L| < ε for all
m, n > N (ε) and denoted by limm,n xmn = L. A double sequence is called bounded
if there exists a positive number M such that |xmn | ≤ M for all (m, n) ∈ N × N. A
real double sequence {xmn }m,n∈N is statistically convergent to L if for every ε > 0,
lim
j,k
|{m ≤ j, n ≤ k : |xmn − L| ≥ ε}|
=0
jk
[27,28].
Recall that a family I ⊂ 2Y of subsets of a nonempty set Y is said to be an ideal
in Y if (i)A, B ∈ I implies A ∪ B ∈ I; (ii)A ∈ I, B ⊂ A implies B ∈ I, while an
admissible ideal I of Y further satisfies {x} ∈ I for each x ∈ Y . If I is a non-trivial
proper ideal in Y (i.e. Y ∈
/ I, I 6= {∅}) then the family of sets F (I) = {M ⊂ Y :
there exists A ∈ I : M = Y \A} is a filter in Y . It is called the filter associated with
the ideal I. A non-trivial ideal I of N × N is called strongly admissible if {i} × N
and N×{i} belong to I for each i ∈ N. It is evident that a strongly admissible ideal
is admissible also. Let I0 = {A ⊂ N × N :3 m(A) ∈ N such that i, j ≥ m(A) =⇒
(i, j) ∈
/ A}. Then I0 is a non-trivial strongly admissible ideal [14]. Let A = (ank )
be a non-negative regular matrix. For an ideal I of N a sequence {xn }n∈N is said
to be AI -statistically convergent to L if for any ε > 0 and δ > 0,
½
¾
X
n∈N:
ank ≥ δ ∈ I,
k∈K(ε)
where K(ε) = {k ∈ N : |xk − L| ≥ ε} [33,34].
Let A = (ajkmn ) be a four dimensional summability matrix. For a given double
sequence {xmn }m,n∈N , the A-transform of x, denoted by Ax := ((Ax)jk ), is given
by
X
(Ax)jk =
ajkmn xmn
(m,n)∈N2
provided the double series converges in Pringsheim sense for every (j, k) ∈ N2 .
In 1926, Robison [30] presented a four dimensional analog of the regularity by
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S. Dutta, P. Das
considering an additional assumption of boundedness. This assumption was made
because a convergent double sequence is not necessarily bounded.
Recall that a four dimensional matrix A = (ajkmn ) is said to be RH-regular
if it maps every bounded convergent double sequence into a convergent double
sequence with the same limit. The Robison-Hamilton conditions state that a four
dimensional matrix A = (ajkmn ) is RH-regular if and only if
(i) limj,k ajkmn = 0 for each (m, n) ∈ N2 ,
P
(ii) limj,k (m,n)∈N2 ajkmn = 1,
P
(iii) limj,k m∈N |ajkmn | = 0 for each n ∈ N,
P
(iv) limj,k n∈N |ajkmn | = 0 for each m ∈ N,
P
(v)
(m,n)∈N2 |ajkmn | is convergent,
P
(vi) there exist finite positive integers M0 and N0 such that m,n>N0 |ajkmn | < M0
holds for every (j, k) ∈ N2 .
Let A = (ajkmn ) be a non-negative RH-regular summability matrix and let
K ⊂ N2 . Then the A-density of K is given by
X
(2)
δA {K} = lim
ajkmn
j,k
(m,n)∈K
provided the limit exists. A real double sequence x = {xmn }m,n∈N is said to be
A-statistically convergent to a number L if for every ε > 0
(2)
δA {(m, n) ∈ N2 : |xmn − L| ≥ ε} = 0.
n
o
(2)
We denote Iδ(2) = C ⊂ N2 : δA {C} = 0 which is an admissible ideal in N × N.
A
Throughout we use I as a non-trivial strongly admissible ideal on N × N.
2. A Korovkin type approximation theorem
Recently the concept of I-statistical convergence for real single sequences has
been introduced by Das and Savas as a notion of convergence which is strictly
weaker than the notion of statistical convergence (see [32] for details). Consequently
this notion has been further investigated in [13]. Very recently it has been further
generalized by using a summability matrix A into AI -statistical convergence for
real single sequences by Savas, Das and Dutta [33,34]. In this paper we consider
the following natural extension of these convergence for real double sequences.
The following definition is due to E. Savas (who has informed about it in a
personal communication).
Definition 2.1. A real double sequence {xm,n }m,n∈N is said to be I2 statistically convergent to L if for each ε > 0 and δ > 0,
n
o
1
(j, k) ∈ N2 : jk
|{m ≤ j, n ≤ k : |xmn − L| ≥ ε}| ≥ δ ∈ I.
We now introduce the main definition of this paper.
Korovkin type approximation
291
Definition 2.2. Let A = (ajkmn ) be a non-negative RH-regular summability
matrix. Then a real double sequence {xmn }m,n∈N is said to be AI2 -statistically
convergent to a number L if for every ε > 0 and δ > 0,
¾
½
X
2
(j, k) ∈ N :
ajkmn ≥ δ ∈ I,
(m,n)∈K2 (ε)
2
where K2 (ε) = {(m, n) ∈ N : |xmn − L| ≥ ε}. In this case, we write AI2 -stlimm,n xmn = L.
It should be noted that, if we take A = C(1, 1), the double Cesáro matrix [26]
defined as follows
½ 1
for m ≤ j, n ≤ k;
ajkmn = jk
0
otherwise,
I
then A2 -statistical convergence coincides with the notion of I2 -statistical convergence. Again if we replace the matrix A by the identity matrix for four dimensional
matrices and I = I0 then AI2 -statistical convergence reduces to the Pringsheim
convergence for double sequences. For the ideal I = I0 , AI2 -statistical convergence
implies A-statistical convergence for double sequences. The basic properties of AI2 statistically convergent double sequences are similar to AI -statistical convergent
single sequences and can be obtained analogously as in [32,33]. So our main aim
here is to present an application of this notion in approximation theory.
Throughout this section, let K = [0, A] × [0, B] A, B ∈ (0, 1) and denote the
space of all real valued continuous functions on K by C(K). This space is endowed
with the supremum norm kf k = sup(x,y)∈K |f (x, y)|, f ∈ C(K). Consider the space
Hw (K) of real valued functions f on K satisfying
³ r u
x 2
v
y 2´
|f (u, v) − f (x, y)| ≤ w f ; (
−
) +(
−
)
1−u 1−x
1−v 1−y
where w is the modulus of continuity for δ > 0 given by
p
w(f ; δ) = sup{|f (u, v) − f (x, y)| : (u, v), (x, y) ∈ K,
(u − x)2 + (v − y)2 ≤ δ}.
Then it is clear that any function in Hw (K) is continuous and bounded on K.
y
x
We will use the following test functions f0 (x, y) = 1, f1 (x, y) = 1−x
, f2 = 1−y
,
y 2
x 2
f3 (x, y) = ( 1−x ) + ( 1−y ) and we denote the value of T f at a point (u, v) ∈ K by
T (f ; u, v).
Now we establish the Korovkin type approximation theorem in AI2 -statistical
sense.
Theorem 2.1. Let {Tmn }m,n∈N be a sequence of positive linear operators from
Hw (K) into C(K) and let A = (ajkmn ) be a non-negative RH-regular summability
matrix. Then for any f ∈ Hw (K),
AI2 -st-lim kTmn f − f k = 0
m,n
is satisfied if the following holds
AI2 -st-lim kTmn fi − fi k = 0, i = 0, 1, 2, 3.
m,n
(1)
(2)
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S. Dutta, P. Das
Proof. Assume that (2) holds. Let f ∈ Hw (K). Our objective is to show that
for given ε > 0 there exist constants C0 , C1 , C2 , C3 (depending on ε > 0) such
that
kTmn f − f k ≤ ε + C3 kTmn f3 − f3 k + C2 kTmn f2 − f2 k
+ C1 kTmn f1 − f1 k + C0 kTmn f0 − f0 k.
If this is done then our hypothesis implies that for any ε > 0, δ > 0,
½
¾
X
(j, k) ∈ N2 :
ajkmn ≥ δ ∈ I,
(m,n)∈K2 (ε)
where K2 (ε) = {(m, n) ∈ N2 : kTmn f − f k ≥ ε}.
To this end, start by observing that for each (u, v) ∈ K the function 0 ≤
s
u 2
t
v 2
guv ∈ Hw (K) defined by guv (s, t) = ( 1−s
− 1−u
) + ( 1−t
− 1−v
) satisfies guv =
y 2
y
x
x 2
2u
2v
u 2
v 2
( 1−x ) +( 1−y ) − 1−u 1−x − 1−v 1−y +( 1−u ) +( 1−v ) . Since each Tmn is a positive
operator, Tmn guv is a positive function. In particular, we have for each (u, v) ∈ K,
0 ≤ Tmn guv (u, v)
y 2
x 2
= [Tmn (( 1−x
) + ( 1−y
) −
y
2v
u 2
v 2
1−v 1−y + ( 1−u ) + ( 1−v ) ; u, v)]
y 2
x 2
u 2
v 2
= [Tmn (( 1−x
) + ( 1−y
) ; u, v) − ( 1−u
) − ( 1−v
) ]
y
2u
x
u
2v
v
− 1−u [Tmn ( 1−x ; u, v) − 1−u ] − 1−v [Tmn ( 1−y ; u, v) − 1−v
]
u 2
v 2
+ {( 1−u
) + ( 1−v
) }[Tmn f0 − f0 ]
2u
≤ kTmn f3 − f3 k + 1−u kTmn f1 − f1 k
u 2
v 2
2v
kTmn f2 − f2 k + {( 1−u
) + ( 1−v
) }kTmn f0 − f0 k.
+ 1−v
2u
x
1−u 1−x
−
Let M = kf k and ε > 0. By the uniform continuity of f on K there exists a
δ > 0 such that −ε < f (s, t) − f (u, v) < ε holds whenever
r
u 2
t
v 2
s
−
) +(
−
) < δ,
(
1−s 1−u
1−t 1−v
(s, t), (u, v) ∈ K. Next observe that
(µ
¶2 µ
¶2 )
2M
s
u
t
v
−ε − 2
−
+
−
δ
1−s 1−u
1−t 1−v
≤ f (s, t) − f (u, v)
(µ
¶2 µ
¶2 )
2M
s
u
t
v
≤ε+ 2
−
+
−
δ
1−s 1−u
1−t 1−v
q
Indeed, if
s
−
( 1−s
u 2
1−u )
t
+ ( 1−t
−
v 2
1−v )
< δ then (3) follows from
−ε < f (s, t) − f (u, v) < ε.
(3)
293
Korovkin type approximation
q
s
u 2
t
v 2
( 1−s
− 1−u
) + ( 1−t
− 1−v
) ≥ δ then (3) follows from
(µ
¶2 µ
¶2 )
2M
s
u
t
v
−ε − 2
−
+
−
δ
1−s 1−u
1−t 1−v
On the other hand, if
≤ −2M ≤ f (s, t) − f (u, v) ≤ 2M
(µ
¶2 µ
¶2 )
s
2M
u
t
v
≤ε+ 2
−
+
−
.
δ
1−s 1−u
1−t 1−v
Since each Tmn is positive and linear it follows from (3) that
2M
2M
−εTmn f0 − 2 Tmn guv ≤ Tmn f − f (u, v)Tmn f0 ≤ εTmn f0 + 2 Tmn guv .
δ
δ
Therefore
|Tmn (f ; u, v) − f (u, v)Tmn (f0 ; u, v)|
≤ ε + ε [Tmn (f0 ; u, v) − f0 (u, v)] +
≤ ε + εkTmn f0 − f0 k +
2M
Tmn guv
δ2
2M
Tmn guv
δ2
In particular, note that
|Tmn (f ; u, v) − f (u, v)|
≤ |Tmn (f ; u, v) − f (u, v)Tmn (f0 ; u, v)| + |f (u, v)| |Tmn (f0 ; u, v) − f0 (u, v)|
2M
≤ ε + (M + ε)kTmn f0 − f0 k + 2 Tmn guv
δ
which implies
kTmn f − f k ≤ ε + C3 kTmn f3 − f3 k + C2 kTmn f2 − f2 k
+ C1 kTmn f1 − f1 k + C0 kTmn f0 − f0 k,
i
A 2
B
A
2
{(
)
+
(
)
}
+
M
+
ε
, C1 = 4M
where C0 = 2M
δ2
1−A
1−B
δ 2 1−A , C2 =
h
C3 =
2M
δ2 ,
i.e.,
kTmn f − f k ≤ ε + C
3
X
kTmn fi − fi k, i = 0, 1, 2, 3,
i=0
where C = max{C0 , C1 , C2 , C3 }.
For a given γ > 0, choose ε > 0 such that ε < γ. Now let
U = {(m, n) : kTmn f − f k ≥ γ}
and
½
Ui =
It follows that U ⊂
γ−ε
(m, n) : kTmn fi − fi k ≥
4C
S3
i=0
¾
, i = 0, 1, 2, 3.
Ui and consequently for all (j, k) ∈ N2
X
(m,n)∈U
ajkmn ≤
3
X
X
i=0 (m,n)∈Ui
ajkmn ,
4M B
δ 2 1−B
and
294
S. Dutta, P. Das
which implies that for any σ > 0 and (m, n) ∈ U ,
½
¾ [
3 ½
X
2
(j, k) ∈ N :
ajkmn ≥ σ ⊆
(j, k) ∈ N2 :
(m,n)∈U
i=0
Therefore from hypothesis, {(j, k) ∈ N2 :
pletes the proof of the theorem.
X
ajkmn
(m,n)∈Ui
P
(m,n)∈U
¾
σ
≥
.
3
ajkmn ≥ σ} ∈ I. This com-
We now show that our theorem is stronger than the A-statistical version [12]
(and so the classical version). Let I be a non-trivial strongly admissible ideal of
N × N. Choose an infinite subset C = {(pi , qi ) : i ∈ N}, from I such that pi 6= qi
for all i, p1 < p2 < · · · and q1 < q2 < · · · . Let {umn }m,n∈N be given by
½
1 m, n are even
umn =
0 otherwise.
Let A = (ajkmn ) be given by


 1 if j = pi , k = qi , m = 2pi , n = 2qi for some i ∈ N
ajkmn = 1 if (j, k) 6= (pi , qi ), for any i, m = 2j + 1, n = 2k + 1


0 otherwise.
Now for 0 < ε < 1, K2 (ε) = {(m, n) ∈ N × N : |umn − 0| ≥ ε} = {(m, n) :
m, n are even}. Observe that
½
X
1 if j = pi , k = qi for some i ∈ N
ajkmn =
0 if (j, k) 6= (pi , qi ), for any i ∈ N.
(m,n)∈K (ε)
2
Thus for any δ > 0,
½
(j, k) ∈ N × N :
X
¾
ajkmn ≥ δ
= C ∈ I,
(m,n)∈K2 (ε)
which shows that {umn }m,n∈N is AI2 -statistically convergent to 0. Evidently this
sequence is not A-statistically convergent to 0.
Consider the following Meyer-König and Zeler operators
Mmn (f ; x, y) = (1 − x)m+1 (1 − y)n+1
µ
¶µ
¶µ
¶
∞ X
∞
X
k
l
m+k
n+l k l
×
f
,
x y
k+m+1 l+m+1
k
l
k=0 l=0
where f ∈ Hw (K) and K = [0, A] × [0, B], A, B ∈ (0, 1). Then Mmn (f0 ; x, y) =
y
x
f0 (x, y), Mmn (f1 ; x, y) = 1−x
, Mmn (f2 ; x, y) = 1−y
and
µ
¶2
µ
¶2
m+2
x
x
n+2
y
y
1
1
Mmn (f3 ; x, y) =
+
.
+
+
m+1 1−x
m+11−x n+1 1−y
n+11−y
Then limm,n kMmn f − f k = 0.
Korovkin type approximation
295
Now consider the following positive linear operator Tmn on Hw (K) defined by
Tmn (f ; x, y) = (1 + umn )Mmn (f ; x, y). It is easy to observe that kTmn fi − fi k =
umn for i = 0, 1, 2 which imply that AI2 -st-limm,n kTmn fi − fi k = 0, i = 0, 1, 2.
Again,
°
n m + 2 ³ x ´2
1
x
n + 2 ³ y ´2
°
kTmn f3 − f3 k = °(1 + umn )
+
+
m+1 1−x
m+11−x n+1 1−y
1
y o ³ x ´2 ³ y ´2 °
°
−
−
+
°
n+11−y
1−x
1−y
½
¾
2
2
m+3
n+3
≤D
+
+ umn
+ umn
,
m+1 n+1
m+1
n+1
o
n
B
A
A
A 2
) , ( 1−B
)2 , ( 1−A
), ( 1−A
) . Therefore
where D = max ( 1−A
©
ª
(m, n) ∈ N2 : kTmn (f3 ) − f3 k ≥ ε
½
¾
1
ε
1
⊆ (m, n) ∈ N2 :
+
≥
m+1 n+1
4D
½
¾
m+3
n+3
ε
∪ (m, n) ∈ N2 : umn
+ umn
≥
m+1
n+1
2D
½
¾
1
1
ε
⊆ (m, n) ∈ N2 :
+
≥
m+1 n+1
4D
r
½
¾
m+3 n+3
ε
∪ (m, n) ∈ N2 : umn +
+
≥2
m+1 n+1
2D
r
½
¾ ½
¾
1
1
ε
ε
2
2
⊆ (m, n) ∈ N :
+
≥
∪ (m, n) ∈ N : umn ≥
m+1 n+1
4D
2D
r
½
¾
m+3 n+3
ε
∪ (m, n) ∈ N2 :
+
≥
m+1 n+1
2D
r
½
¾ ½
¾
1
ε
1
ε
2
2
⊆ (m, n) ∈ N :
+
≥
∪ (m, n) ∈ N : umn ≥
m+1 n+1
4D
2D
r
½
¾
1
1
1
ε
∪ (m, n) ∈ N2 :
+
≥
m+1 n+1
6 2D
r
½
¾
m
n
1
ε
2
∪ (m, n) ∈ N :
+
≥
,
m+1 n+1
2 2D
which implies that AI2 -st-limm,n kTmn f3 − f3 k = 0. Hence from previous theorem
it follows that AI2 -st-limm,n kTmn f − f k = 0 for any f ∈ Hw (K). But since
{umn }m,n∈N is not A-statistically convergent so the sequence {Tmn (f ; x, y)}m,n∈N
considered above does not converge A-statistically to the function f ∈ Hw (K).
3. Rate of AI2 -statistical convergence
In this section we present a way to compute the rate of AI2 -statistical convergence in Theorem 2.1. We will need the following definitions.
296
S. Dutta, P. Das
Definition 3.1. Let A = (ajkmn ) be a non-negative RH-regular summability
matrix and let {αmn }m,n∈N be a positive non-increasing double sequence. Then
a real double sequence {xmn }m,n∈N is said to be AI2 -statistically convergent to a
number L with the rate of o(αmn ) if for every ε > 0 and δ > 0,
¾
½
X
1
2
(j, k) ∈ N :
ajkmn ≥ δ ∈ I,
αjk
(m,n)∈K2 (ε)
where K2 (ε) = {(m, n) ∈ N
AI2 -st-o(αmn )-limm,n xmn = L.
2
:
|xmn − L| ≥ ε}.
In this case, we write
Definition 3.2. Let A = (ajkmn ) be a non-negative RH-regular summability
matrix and let {αmn }m,n∈N be a positive non-increasing double sequence. Then
a real double sequence {xmn }m,n∈N is said to be AI2 -statistically convergent to a
number L with the rate of om (αmn ) if for every ε > 0 and δ > 0,
½
¾
X
2
(j, k) ∈ N :
ajkmn ≥ δ ∈ I,
(m,n)∈K2 (ε)
2
where K2 (ε) = {(m, n) ∈ N :
AI2 -st-om (αmn )-limm,n xmn = L.
|xmn − L| ≥ εαmn }. In this case, we write
Lemma 3.1. Let {xmn }m,n∈N and {ymn }m,n∈N be double sequences. Assume that A = (ajkmn ) is a non-negative RH-regular summability matrix and let
{αmn }m,n∈N and {βmn }m,n∈N be positive non-increasing double sequences. If
AI2 -st-o(αmn )-lim xmn = L1 and AI2 -st-o(βmn )-lim xmn = L2
m,n
m,n
then we have
(i) AI2 -st-o(γmn )-lim (xmn ± ymn ) = L1 ± L2 where γmn = max{αmn , βmn },
m,n
(ii) AI2 -st-o(αmn )-lim λxmn = λL1 for any real number λ.
m,n
Proof. The proof is straightforward and so is omitted.
Lemma 3.2. Let {xmn }m,n∈N and {ymn }m,n∈N be double sequences. Assume that A = (ajkmn ) is a non-negative RH-regular summability matrix and let
{αmn }m,n∈N and {βmn }m,n∈N be positive non-increasing double sequences. If
AI2 -st-om (αmn )-lim xmn = L1 and AI2 -st-om (βmn )-lim xmn = L2
m,n
m,n
then we have
(i) AI2 -st-om (γmn )-lim (xmn ± ymn ) = L1 ± L2 where γmn = max{αmn , βmn },
m,n
(ii)
AI2 -st-om (αmn )-lim
m,n
λxmn = λL1 for any real number λ.
297
Korovkin type approximation
Proof. The proof is straightforward and so is omitted.
Now we prove the following theorem.
Theorem 3.1. Let {Tmn }m,n∈N be a sequence of positive linear operators from
Hw (K) into C(K). Let A = (ajkmn ) be a non-negative RH-regular summability matrix and {αmn }m,n∈N and {βmn }m,n∈N be positive non-increasing double sequences.
Assume that the following conditions hold
(i) AI2 -st-o(αmn )-lim kTmn f0 − f0 k = 0,
m,n
(ii)
p
AI2 -st-o(βmn )-lim
m,n
w(f ; δmn ) = 0,
x
−
where δ := δmn = kTmn (ψ)k with ψ(u, v) = ( 1−x
for any f ∈ Hw (K),
u 2
1−u )
y
+ ( 1−y
−
v 2
1−v ) .
Then
AI2 -st-o(γmn )-lim kTmn f − f k = 0,
m,n
where γmn = max{αmn , βmn } for each (m, n) ∈ N2 .
Proof. Let {Tmn }m,n∈N be a sequence of positive linear operators from Hw (K)
into C(K) and let A = (ajkmn ) be a non-negative RH-regular summability matrix
and N = kf k. Then for any f ∈ Hw (K),
|Tmn (f ; u, v) − f (u, v)|
≤ Tmn (|f (x, y) − f (u, v)|; u, v) + |f (u, v)kTmn (f0 ; u, v) − f0 (u, v)|
q


y 2
u
x 2
v
( 1−u
− 1−x
) + ( 1−v
− 1−y
)
≤ w(f ; δ)Tmn 1 +
; u, v 
δ
+ N |Tmn (f0 ; u, v) − f0 (u, v)|
q
= w(f ; δ)Tmn (f0 ; u, v) + w(f ; δ)Tmn 
u
( 1−u
−
x 2
1−x )
v
+ ( 1−v
−
y 2
1−y )
δ

; u, v 
+ N |Tmn (f0 ; u, v) − f0 (u, v)|
= w(f ; δ)Tmn (f0 ; u, v) − w(f ; δ)f0 (u, v) + w(f ; δ) +
w(f ; δ)
Tmn (ψ; u, v)
δ2
+ N |Tmn (f0 ; u, v) − f0 (u, v)|
≤ w(f ; δ)|Tmn (f0 ; u, v) − f0 (u, v)| + w(f ; δ) +
w(f ; δ)
Tmn (ψ; u, v)
δ2
+ N |Tmn (f0 ; u, v) − f0 (u, v)|.
Taking supremum over (u, v) ∈ K,
w(f ; δ)
kTmn ψk + N kTmn f0 − f0 k.
kTmn f − f k ≤ w(f ; δ)kTmn f0 − f0 k + w(f ; δ) +
δ2
p
If we take δ := δmn = kTmn ψk then
kTmn f − f k ≤ w(f ; δ)kTmn f0 − f0 k + 2w(f ; δ) + N kTmn f0 − f0 k
≤ M {w(f ; δ)kTmn f0 − f0 k + w(f ; δ) + kTmn f0 − f0 k},
298
S. Dutta, P. Das
where M = max{2, N }. Let µ > 0 be given. Now consider the following sets
U = {(m, n) : kTmn f − f k ≥ µ},
µ
U1 = {(m, n) : w(f ; δ) ≥
},
3M
µ
U2 = {(m, n) : kTmn f0 − f0 k ≥
},
3M
U3 = {(m, n) : w(f ; δ)kTmn f0 − f0 k ≥
Then U ⊂ U1 ∪ U2 ∪ U3 . Now define
µ
}.
3M
r
µ
},
3M
r
µ
00
U3 = {(m, n) : kTmn f0 − f0 k ≥
}.
3M
U30
= {(m, n) : w(f ; δ) ≥
Then U ⊂ U1 ∪U2 ∪U30 ∪U300 . Now since γmn = max{αmn , βmn } for each (m, n) ∈ N2
then for all (j, k) ∈ N2 ,
X
X
X
1
1
1
ajkmn ≤
ajkmn +
ajkmn
γj,k
βj,k
αj,k
(m,n)∈U
(m,n)∈U1
1
+
βj,k
X
(m,n)∈U2
1
ajkmn +
α
j,k
0
(m,n)∈U3
X
ajkmn .
(m,n)∈U300
Then for any σ > 0
¾
½
X
1
ajkmn ≥ σ
(j, k) ∈ N2 :
γj,k
(m,n)∈U
½
¾ ½
X
X
1
σ
1
2
⊆ (j, k) ∈ N :
ajkmn ≥
∪ (j, k) ∈ N2 :
ajkmn ≥
βj,k
4
αj,k
(m,n)∈U1
(m,n)∈U2
½
¾ ½
X
X
1
σ
1
∪ (j, k) ∈ N2 :
ajkmn ≥
∪ (j, k) ∈ N2 :
ajkmn ≥
βj,k
4
αj,k
0
00
(m,n)∈U3
(m,n)∈U3
σ
4
¾
¾
σ
.
4
Now from hypothesis the sets on the right-hand side belong to I and consequently
¾
½
X
1
2
ajkmn ≥ σ ∈ I
(j, k) ∈ N :
γj,k
(m,n)∈U
for any σ > 0. This completes the proof.
The proof of the following theorem is analogous to the proof of Theorem 3.1
and so is omitted.
Theorem 3.2. Let {Tmn }m,n∈N be a sequence of positive linear operators from
Hw (K) into C(K). Let A = (ajkmn ) be a non-negative RH-regular summability matrix and {αmn }m,n∈N and {βmn }m,n∈N be positive non-increasing double sequences.
299
Korovkin type approximation
Assume that the following conditions hold
(i) AI2 -st-om (αmn )-lim kTmn f0 − f0 k = 0,
m,n
(ii)
AI2 -st-om (βmn )-lim
m,n
w(f ; δmn ) = 0,
p
y
x
u 2
where δmn = kTmn (ψ)k with ψ(u, v) = ( 1−x
− 1−u
) + ( 1−y
−
any f ∈ Hw (K),
AI2 -st-om (γmn )-lim kTmn (f ) − f k = 0,
v 2
1−v ) .
Then for
m,n
where γmn = max{αmn , βmn } for each (m, n) ∈ N2 .
Acknowledgement. The first author is thankful to CSIR, India for giving
JRF during the tenure of which this work was done.
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(received 03.11.2014; in revised form 29.05.2015; available online 20.06.2015)
Department of Mathematics, Jadavpur University, Jadavpur, Kol-32, West Bengal, India
E-mail: dutta.sudipta@ymail.com, sudiptaju.scholar@gmail.com,
pratulananda@yahoo.co.in