Korovkin type approximation theorem for functions of two variables through statistical A-summability

Mursaleen Mohammad, Alotaibi, Alotaibi Abdullah

Description

In this article, we prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical A -summability. We also study the rate of statistical A -summability of positive linear operators. Finally we construct an example by Bleimann et al. operators to show that our result is stronger than those of previously proved by other authors. AMS Subject Classification 2000: 41A10; 41A25; 41A36; 40A30; 40G15.

Sujets

• Density

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 8
• Langue: English

Extrait

Mursaleen and Alotaibi Advances in Difference Equations 2012, 2012:65
http://www.advancesindifferenceequations.com/content/2012/1/65
RESEARCH Open Access
Korovkin type approximation theorem for functions
of two variables through statisticalA-summability
1* 2Mohammad Mursaleen and Abdullah Alotaibi
* Correspondence: Abstract
[e-mail]
1Department of Mathematics, In this article, we prove a Korovkin type approximation theorem for a function of
Aligarh Muslim University, Aligarh two variables by using the notion of statistical A-summability. We also study the rate
202002, India
of statistical A-summability of positive linear operators. Finally we construct anFull list of author information is
available at the end of the article example by Bleimann et al. operators to show that our result is stronger than those
of previously proved by other authors.
AMS Subject Classification 2000: 41A10; 41A25; 41A36; 40A30; 40G15.
Keywords: density, statistical convergence, A-statistical convergence, statistical A-
summability, positive linear operator, Korovkin type approximation theorem
1 Introduction and preliminaries
The concept of statistical convergence for sequences of real numbers was introduced
by Fast [1] and further studied many others.
Let K⊆N and K ={k≤ n : kÎ K}. Then the natural density of K is defined by δ(K)=n
-1
lim n |K | if the limit exists, where |K | denotes the cardinality of K .n n n n
A sequence x=(x ) of real numbers is said to be statistically convergent to L providedk
that for every ε >0 the set K :={kÎN:|x -L|≥ε} has natural density zero, i.e. for each ε>0,ε k
1
lim |{k ≤ n : |x −L|≥ ε}| =0.k
n n
In this case we write st- lim x = L. Note that if x=(x ) is convergent then it is statis-k
tically convergent but not conversely. The idea of statistical convergence of double
sequences has been intruduced and studied in [2,3].
Let A=(a ), n, kÎN,beaninfinitematrixand x=(x ) be a sequence. Then thenk k
(transformed) sequence, Ax := (y ), is denoted byn
∞
y := a x ,n nk k
k=1
where it is assumed that the series on the right converges for each nÎN. We say that
a sequence x is A-summable to the limit ℓ if y ® ℓ as n® ∞.n
A matrix transformation is said to be regular if it maps every convergent sequence
into a convergent sequence with the same limit. The well-known conditions for two
dimensional matrix to be regular are known as Silverman-Toeplitz conditions.
© 2012 Mursaleen and Alotaibi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.Mursaleen and Alotaibi Advances in Difference Equations 2012, 2012:65 Page 2 of 10
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In [4], Edely and Mursaleen have given the notion of statistical A-summability for
single sequences and statistical A-summability for double sequences has recently been
studied in [5].
Let A=(a ) be a nonnegative regular summability matrix and x=(x)beank k
sequence of real or complex sequences. We say that x is statistically A-summable to L
if for every ε >0,

δ n ∈N : y −L ≥ ε =0.n
So, if x is statistically A-summable to L then for every ε >0,
1
lim n ≤ m : y −L ≥ ε =0.n
m m
Note that if a sequence is bounded and A-statistically convergent to L,thenitis A-
summable to L; hence it is statistically A-summable to L but not conversely (see [4]).
Example 1.1. Let A=(C, 1), the Cesàro matrix and the sequence u=(u ) be definedk
by

1 ifk is odd,
u = (1:1)k 0 ifk is even.
1Then u is is A-summable to (and hence statistically A-summable to 1/2) but not2
statistically (and not A-statistically as well) convergent.
Let I := [0, ∞) and C(I) denote the space of all continuous real valued functions on I.
Let C (I):= {fÎ C(I): f is bounded on I}.C(I) and C (I) are equipped with normB B
||f|| := sup |f(x)|.C(I)
x∈I
Let H (I) denote the space of all real valued functions f on I such thatω

s x
|f(s) −f(x)|≤ ω f; − ,
1+s 1+x
where ω is the modulus of continuity, i.e.
ω(f;δ)=sup{|f(s) −f(x)| : |s −x|≤ δ}.
s,x∈I
It is to be noted that any function fÎ H (I) is continuous and bounded on I.ω
The following Korovkin type theorem (see [6]) was proved by Çakar and Gadjiev [7].
Theorem A.Let(L ) be a sequence of positive linear operators from H (I)into Cn ω B
(I). Then for all fÎ H (I)ω

lim L (f;x) −f(x) = 0n C (I)Bn→∞
if and only if

lim L (f ;x) −g =0(i=0,1,2),n i i C (I)Bn→∞
where

2x x
g (x)=1,g (x)= ,g (x)= .0 1 2
1+x 1+x
Mursaleen and Alotaibi Advances in Difference Equations 2012, 2012:65 Page 3 of 10
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Erkuş and Duman [8] have given the st -version of the above theorem for functionsA
of two variables.Quite recently, Korovkin type of approximation theorems have been
proved in [9,10] by using almost convergence; in [11-15] by using variants of statistical
convergence and in [16-19] for functions of two variables by using statistical conver-
gence, A-statistical convergence and statistical A-summability of double sequences. In
this article, we use the notion of statistical A-summability to prove a Korovkin type
approximation theorem for functions of two variables with the help of test functions
y 2 y 2x x1, , ,( ) +( ) .1+x 1+y 1+x 1+y
2 Main result
Let I = [0, ∞) and K = I × I. We denote by C (K) the space of all bounded and contin-B
uous real valued functions on K equipped with norm
||f|| := sup |f(x,y)|, f ∈ C (K).C (K) BB
(x,y)∈K
Let H (K) denote the space of all real valued functions f on K such thatω*
⎛ ⎞
22s x t y∗⎝ ⎠|f(s,t) −f(x,y)|≤ ω f; − + − ,
1+s 1+x 1+t 1+y
where ω* is the modulus of continuity, i.e.

∗ 2 2ω (f;δ)= sup |f(s,t) −f(x,y)| : (s −x) +(t −y) ≤ δ .
(s,t),(x,y)∈K
It is to be noted that any function fÎ H (K) is bounded and continuous on K,andω*
a necessary and sufficient condition for fÎ H (K) is thatω*
∗lim ω (f;δ)=0.
δ→0
We prove the following result:
Theorem 2.1. Let A=(a ) be nonnegative regular summability matrix. Let (T)beank k
sequence of positive linear operators from H (K) into C (K). Then for all fÎ H (K)ω* B ω*

∞
st - lim a T (f;x,y) −f(x,y) = 0nk k (2:0)
n→∞
k=1 C (K)B
if and only if

∞
st - lim a T (1;x,y) −1 = 0 nk k (2:1)
n→∞
k=1 C (K)B

∞
s x
st - lim a T ;x,y − = 0nk k (2:2)
n→∞ 1+s 1+x
k=1 C (K)B
∞ t y
st - lim a T ;x,y − = 0 nk k (2:3)
n→∞ 1+t 1+y
k=1 C (K)BMursaleen and Alotaibi Advances in Difference Equations 2012, 2012:65 Page 4 of 10
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∞ 2 2

2 2s t x y
st - lim a T + ;x,y − + =0.nk k (2:4)
n→∞ 1+s 1+t 1+x 1+y
k=1 C (K)B
Proof. Since each of the functions

2 2y yx x belongs to H (K),f (x,y)=1,f (x,y)= ,f (x,y)= ,f (x,y)= +0 1 2 3 ω*1+x 1+y 1+x 1+y
conditions (2.1)-(2.4) follow immediately from (2.0).
Let fÎ H (K) and (x, y)Î K be fixed. Then for ε > 0 there exist δ , δ >0 such thatω* 1 2
ys x t| − | <δ|f(s, t)-f(x, y)| <ε holds for all (s, t)Î K satisfying | − | <δ and .1 21+s 1+x 1+t 1+y
Let
⎧ ⎫
⎨ 2 ⎬
2s x t y
K(δ):= (s,t) ∈ K : − + − <δ = min{δ ,δ } .1 2
⎩ 1+s 1+x 1+t 1+y ⎭
Hence
|f(s,t) −f(x,y)| = |f(s,t) −f(x,y)|χ (s,t)+ |f(s,t) −f(x,y)|χ (s,t)K(δ) K\K(δ)
(2:5)
≤ ε+2Nχ (s,t)K\K(δ)
N = ||f||Wherec denotes the characteristic function of the set D and . FurtherD C (K)B
we get
2
21 s x 1 t y
(2:6)χ (s,t) ≤ − + − .K\K(δ) 2 2δ 1+s 1+x δ 1+t 1+y1 2
Combining (2.5) and (2.6), we get
2
22N s x t y
|f(s,t) −f(x,y)|≤ ε + − + − , (2:7)
2δ 1+s 1+x 1+t 1+y
After using the properties of f, a simple calculation gives that
|T (f;x,y) −f(x,y)|≤ ε +M{|T (f ;x,y) −f (x,y)|+ |T (f ;x,y) −f (x,y)|k k 0 0 k 1 1
(2:8)
+ |T (f ;x,y) −f (x,y)|+ |T (f ;x,y) −f (x,y)|},k 2 2 k 3 3
where
4N
M := ε +N + .
2δ
∞Now replacing T (f; x, y)by a T (f;x,y)and taking sup , we getk nk k (x,y)ÎKk=1
⎛
∞ ∞ ⎝ a T (f;x,y) −f(x,y) ≤ ε +M a T (f ;x,y) −f (x,y)nk k nk k 0 0

k=1 k=1C (K) C (K)B B

∞ ∞
+ a T (f ;x,y) −f (x,y) + a T (f ;x,y) −f (x,y) nk k 1 1 nk k 2 2 (2:9)

k=1 k=1C (K) C (K)B B
⎞
∞ ⎠+ a T (f ;x,y) −f (x,y) .nk k 3 3

k=1 C (K)BMursaleen and Alotaibi Advances in Difference Equations 2012, 2012:65 Page 5 of 10
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For a givenr>0 choose ε > 0 such that ε <r. Define the following sets
⎧ ⎫
∞⎨ ⎬
D := n : a T (f;x,y) −f(x,y) ≥ r ,nk k
⎩ ⎭
k=1 C (K)B
⎧ ⎫
∞⎨ ⎬ r − ε
D := n : a T (f ;x,y) −f (x,y) ≥ ,1 nk k 0 0
⎩ ⎭4K
k=1 C (K)B⎧ ⎫
∞⎨ ⎬ r − ε
D := n : a T (f ;x,y) −f (x,y) ≥ , 2 nk k 1 1⎩ ⎭4K
k=1 C (K)B⎧ ⎫
∞⎨ ⎬ r − ε
D := n : a T (f ;x,y) −f (x,y) ≥ , 3 nk k 2 2
⎩ 4K ⎭
k=1 C (K)B
⎧ ⎫
∞⎨ ⎬ r − ε <