JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 491768 10.1155/2013/491768 491768 Research Article Statistical Approximation for Periodic Functions of Two Variables Alotaibi Abdullah 1 Mursaleen M. 2 Mohiuddine S. A. 1 Aghajani Asadollah 1 Department of Mathematics Faculty of Science King Abdulaziz University P.O. Box 80203 Jeddah 21589 Saudi Arabia kau.edu.sa 2 Department of Mathematics Aligarh Muslim University Aligarh 202002 India amu.ac.in 2013 28 11 2013 2013 04 10 2013 17 11 2013 2013 Copyright © 2013 Abdullah Alotaibi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability (C,1,1). We also study the rate of statistical summability (C,1,1) of positive linear operators. Finally we construct an example to show that our result is stronger than those previously proved for Pringsheim's convergence and statistical convergence.

1. Introduction and Preliminaries

In 1951, Fast  and Steinhaus  independently introduced an extension of the usual concept of sequential limit which is called statistical convergence.

The number sequence  x  is said to be statistically convergent to the number L provided that for each  ϵ>0, (1)limn1n|{kn;|xk-|ϵ}|=0, where  |{kn:kK}|  denotes the number of elements of  K  not exceeding  n. In this case we write st-limxk=.

The notion of statistical convergence of double sequences  x=(xjk)  has been introduced and studied in [3, 4] independently in the same year, 2003.

Let  K×  be a two-dimensional set of positive integers and let  K(n,m)  be the numbers of  (i,j)  in  K  such that  in  and  jm.  Then the two-dimensional analogue of natural density can be defined as follows.

The lower asymptotic density of a set  K× is defined as (2)δ2(K)=limn,minfK(n,m)nm. In this case the sequence  (K(n,m)/nm)  has a limit in Pringsheim’s sense then we say that  K  has a double natural density and is defined as (3)P-limn,mK(n,m)nm=δ2(K).

A real double sequence  x=(xjk)  is said to be statistically convergent to the number l if for each  ϵ>0, the set (4){(i,j),jm,kn:|xjk-|ϵ} has double natural density zero. In this case we write  st2-limj,kxjk=.

If  x  is statistically convergent, then  x  need not be convergent. Also it is not necessarily bounded. For example, let  x=(xjk)  be defined as (5)xjk={jk,if  j  and  k  are  squares,1,otherwise. It is easy to see that  st2-limxjk=1, since the cardinality of the set  {(j,k):|xjk-1|ϵ}jk  for every  ϵ>0. But  x  is neither convergent nor bounded.

Móricz  introduced the notion of statistical summability  (C,1,1).  A double sequence  x=(xjk)  is said to be statistically summable  (C,1,1)  to the number    if for every  ϵ>0, (6)δ2{(m,n)×:|σmn-|ϵ}=0, where (7)σmn=1(m+1)(n+1)j=0mk=0nxjk is the  (C,1,1)  mean of  x=(xjk).  Thus, the double sequence  x  is statistically summable  (C,1,1)  to l if and only if the sequence  σ=(σmn)  is statistically convergent to . In this case we write st2(C,1,1)-limj,kxjk=. Note that if a double sequence is bounded then  st2-limj,kxjk= implies  st2-limm,nσmn=.

Korovkin type approximation theorems (cf. ) are useful tools to check whether a given sequence  (Ln)n1  of positive linear operators on  C[0,1]  of all continuous functions on the real interval  [0,1]  is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for test functions 1, x, and x2 in the space  C[0,1]  as well as for test functions 1, cos, and sin in the space of all continuous 2π-periodic functions on the real line.

We know that  C[0,1]  is a Banach space with norm (8)f:=supx[0,1]|f(x)|,fC[0,1].

We denote by  C2π([0,1])  the space of all  2π-periodic functions  fC([0,1])  which is a Banach space with (9)f2π=supt[0,1]|f(t)|.

After the paper of Gadjiev and Orhan , many papers have appeared in the literature concerning the Korovkin type approximation theorems via different statistical summability methods and for different sets of test functions. At present we are concerned about applications of such summability methods for double sequences to prove two-dimensional version of Korovkin theorem. For example, in [12, 13] the authors used the notion of statistical  A-summability of double sequences; in , the authors have used, respectively, statistical convergence and  A-statistical convergence of double sequences; and in [17, 18], the authors used almost summability. For some more related work, we refer to .

In this paper, we present the Korovkin type approximation theorem for periodic functions via statistical summability  (C,1,1)  and also study the rate of statistical summability  (C,1,1)  of a double sequence of positive linear operators defined from  C*(2)  into  C*(2),where  C*(2)  is the space of all  2π-periodic and real valued continuous functions on  2  equipped with the norm (10)fC(2):=sup(x,y)2|f(x,y)|,(fC(2)).

2. Main Result

First, we state the result due to  for  A-statistical convergence of double sequences.

Theorem 1.

Let  (Lmn)  be a double sequence of positive linear operators acting from  C*(2)  into  C*(2).  Then, for all  fC*(2)(11)st2A-limm,nLmn(f)-fC*(2)=0 if and only if (12)st2A-limm,nLmn(fi)-fiC*(2)=0,i=0,1,2,3,4, where  f0(x,y)=1,  f1(x,y)=sinx,f2(x,y)=siny,f3(x,y)=cosx, and  f4(x,y)=cosy.

If we replace the matrix  A  by the identity four-dimensional matrix in the above theorem, then we immediately get the following result in Pringsheim’s sense.

Corollary 2.

Let  (Lmn)  be a double sequence of positive linear operators acting from  C*(2)  into  C*(2).  Then, for all  fC*(2)(13)P-limm,nLmn(f)-fC*(2)=0 if and only if (14)P-limm,nLmn(fi)-fiC*(2)=0,i=0,1,2,3,4. We prove the following result.

Theorem 3.

Let  (Tjk)  be a double sequence of positive linear operators acting from  C*(2)  into  C*(2).  Then, for all  fC*(2)(15)st2(C,1,1)-limj,kTjk(f)-fC*(2)=0 if and only if (16)st2(C,1,1)-limj,kTjk(fi)-fiC*(2)=0st2(C,1,1)-  limj,ks(C1)  (i=0,1,2,3,4).

Proof.

Since each of the functions f0, f1, f2, f3, and f4 belongs to C*(2), necessity follows immediately from (15). Let condition (16) hold and  fC*(2).  Let  I  and  J  be closed subintervals each of length 2π of . Fix(x,y)I×J. By the continuity of  f  at  (x,y), it follows that for given ε>0 there is a number  δ>0  such that for all  (u,υ)2(17)|f(u,υ)-f(x,y)|<ε, whenever  |u-x|,|υ-y|<δ.  Since  f  is bounded, it follows that (18)|f(u,υ)-f(x,y)|Mf=fC*(2), for all  (u,υ)2.

For all  (u,υ)(x-δ,2π+x-δ]×(y-δ,2π+y-δ], it is well known that (19)|f(u,υ)-f(x,y)|<ε+2Mfsin2(δ/2)ψ(u,υ), where  ψ(u,υ)=sin2((u-x)/2)+sin2((υ-y)/2).  Since the function  fC*(2)  is  2π-periodic, the inequality (19) holds for  (u,υ)2. Then, we obtain (20)|Tjk(f;x,y)-f(x,y)|Tjk(|f(u,υ)-f(x,y)|;x,y)+|f(x,y)||Tjk(f0;x,y)-f0(x,y)||Tjk(ε+2Mfsin2(δ/2)ψ(u,υ);x,y)|+Mf|Tjk(f0;x,y)-f0(x,y)|ε+((ε+Mf)|Tjk(f0;x,y)-f0(x,y)|)+Mfsin2(δ/2)×{2|Tjk(f0;x,y)-f0(x,y)|gg+|sinx||Tjk(f1;x,y)-f1(x,y)|gg+|siny||Tjk(f2;x,y)-f3(x,y)|gg+|cosx||Tjk(f3;x,y)-f3(x,y)|gg+|cosy||Tjk(f4;x,y)-f4(x,y)|}<ε+Ki=04|Tjk(fi;x,y)-fi(x,y)|, where  K:=ε+Mf+(2Mf/sin2(δ/2)). Now, taking  sup(x,y)I×J, we get (21)Tjk(f)-fC*(2)<ε+Ki=04Tjk(fi)-fiC*(2). Now for a given  r>0  choose  ε>0  such that  ε<r. Define the following sets: (22)D={(m,n):Lmn(f)-fC*(2)r},Di={(m,n):Lmn(fi)-fiC*(2)r-ε5K}ggggggggggg(i=0,1,2,3,4), where  Lmn=(1/(m+1)(n+1))j=0mk=0nTjk. Then by (21) (23)Di=04Di, and so (24)δA(2)(D)i=04δA(2)(Di).

Now using (16), we get (25)st2(C,1,1)-limj,kTjk(f)-fC*(2)=0.

Example 4.

Now we present an example of double sequences of positive linear operators, showing that Corollary 2 does not work but our approximation theorem works. We consider the double sequence of Fejer operators on  C*(2)(26)σmn(f;x,y)=1(nπ)·1(nπ)×-ππ-ππf(u,υ)Fm(u)Fn(υ)dudυ, where (27)Fm(u)=sin2(m(u-x)/2)sin2((u-x)/2),1π-ππFm(u)du=1. Observe that (28)σmn(f0;x,y)=f0(x,y),σmn(f1;x,y)=m-1mf1(x,y),σmn(f2;x,y)=n-1nf2(x,y),σmn(f3;x,y)=m-1mf3(x,y),σmn(f4;x,y)=n-1nf4(x,y). Define a double sequence  α=(αmn)  by  αmn=(-1)m+n, m,n.

We observe that α=(αmn) is neither  P-convergent nor statistically convergent but (29)st2(C,1,1)-limα=0.

Let us define the operators  Lmn:C*(2)C*(2)  by (30)Lmn(f;x,y)=(1+αmn)σmn(f;x,y). Then, observe that the double sequence of positive linear operators  (Lmn)  defined by (30) satisfies all hypotheses of Theorem 3. Hence, by (28), we have, for all  fC*(2), (31)st2(C,1,1)-limm,nLmn(f)-fC*(2)=0. Since  (αmn)  is neither  P-convergent nor statistically convergent, the sequence  (Lmn)  given by (30) is also neither  P-convergent nor statistically convergent to the function  fC*(2). So, we conclude that Corollary 2 and Theorem 1 do not work for the operators  (Lmn)  given by (30) while Theorem 3 still works. Hence, we conclude that  st2(C,1,1)-version is stronger than that of  P-version as well as statistical version.

3. Rate of Statistical Summability <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M167"><mml:mo mathvariant="bold">(</mml:mo><mml:mi>C</mml:mi><mml:mo mathvariant="bold">,</mml:mo><mml:mn>1,1</mml:mn><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>

Let  (βmn)  be a positive nonincreasing double sequence. We say that a double sequence  x=(xmn)  is statistically summable  (C,1,1)  to the number  L  with the rate  o(βmn)  if for every  ε>0, (32)P-limm,n1βmn|{jm,kn:|σjk-|ϵ}|=0. In this case, we write  xmn-L=st2(C,1,1)-o(βmn)  as  m,n.

Now, we recall the notion of modulus of continuity. The modulus of continuity of fC*(2), denoted by ω(f,δ) for δ>0, is defined by (33)ω(f,δ)=sup{(u-x)2+(υ-y)2|f(u,υ)-f(x,y)|:(u,υ),(x,y)2,fjjjj(u-x)2+(υ-y)2δ}. It is well known that (34)|f(u,υ)-f(x,y)|ω(f,(u-x)2+(υ-y)2)ω(f,δ)((u-x)2+(υ-y)2δ+1). Then we have the following result.

Theorem 5.

Let  (Tjk)  be a double sequence of positive linear operators acting from C*(2) into C*(2). Let (αjk) and (βjk) be two positive non-increasing sequences. Suppose that

Tjk(f0)-f0C*(2)=st2(C,1,1)-o(αmn),

ω(f,λjk)=st2(C,1,1)  -  o(βjk), where  λmjk=Tjk(φ)C*(2)  and (35)φ(u,υ)=sin2(u-x2)+sin2(υ-y2)foreach(u,υ),(x,y)2.

Then, for all  fC*(2), (36)Tjk(f)-fC*(2)=st2(C,1,1)-o(γjk), where  γjk=max{αjk,βjk}.

Proof.

Let fC*(2) and (x,y)-[π,π]×[-π,π]. Let  δ>0; we have the following cases.

Case I. If  δ<|u-x|π,  δ<|υ-y|π, then  |u-x|π|sin((u-x)/2)|  and  |υ-y|π|sin((υ-y)/2)|. Therefore by (34), we have (37)|f(u,υ)-f(x,y)|ω(f,δ)(π2sin2((u-x)/2)+sin2((υ-y)/2)δ2+1)  .

Case II. If  |u-x|>π,|υ-y|π. Let  k  be an integer such that  |u+2kπ-x|π; then (38)|f(u,υ)-f(x,y)|=|f(u+2kπ,υ)-f(x,y)|ω(f,δ)×(π2sin2((u+2kπ-x)/2)+sin2((υ-y)/2)δ2+1)=ω(f,δ)(π2sin2((u-x)/2)+sin2((υ-y)/2)δ2+1). Similarly, in the other two cases when  |u-x|π, |υ-y|>π and |u-x|>π, |υ-y|>π, we obtain (37).

Now, using the definition of modulus of continuity and the linearity and the positivity of the operators  Tjk,we get (39)|Tjk(f;x,y)-f(x,y)|Tjk(|f(u,υ)-f(x,y)|;x,y)+|f(x,y)||Tjk(f0;x,y)-f0(x,y)|ω(f,δ)Tjk(f0;x,y)+π2ω(f,δ)δ2Tjk(φ;x,y)+|f(x,y)||Tjk(f0;x,y)-f0(x,y)|. Taking supremum over  (x,y)  on both sides of the above inequality and let (40)δ:=δjk=Tjk(φ)C*(2). We obtain (41)Tjk(f)-fC*(2)ω(f,δjk)Tjk(f0)-f0C*(2)+(1+π2)×ω(f,δjk)+MTjk(f0)-f0C*(2), where  M:=fC*(2). Let Lmn=(1/(m+1)(n+1))j=0mk=0nTjk. Now for a given  ε>0  define the following sets: (42)D={(m,n):Lmn(f)-fC*(2)ε},D1={(m,n):Lmn(f0)-f0C*(2)ε3},D2={(m,n):ω(f,δmn)ε3(1+π2)},D3={(m,n):Lmn(f)-fC*(2)ε3M}. Then  DD1D2D3. Further define (43)D4={(m,n):ω(f,δmn)ε3},D5={(m,n):Lmn(f)-fC*(2)ε3}. We see that  D1D4D5. Therefore  Di=25Di. Therefore, since  γmn=max{αmn,βmn}, we conclude that for every  (j,k)×(44)δ2(D)i=25δ2(Di). Using conditions (i) and (ii), we get  Lmn(f)-fC*(2)=st2(C,1,1)-o(γmn).

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

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