Korovkin type approximation theorem for functions of two variables via statistical summability (C, 1)

The concept of statistical summability (C, 1) has recently been introduced by Moricz (2002). In this paper, we use this notion of summability to prove the Korovkin type approximation theorem for functions of two variables. Finally we construct an example by Bleimann, Butzer and Hahn operators to show that our result is stronger than those of previously proved by other authors for ordinary convergence and statistical convergence.


Introduction
The concept of statistical convergence for sequences of real numbers was introduced by Fast (1951) and further studied by many others (FRIDY 1985;AIYUB, 2012;ALGHAMDI, 2012;MOHIUDDINE et al., 2010 is convergent then it is statisically convergent but not conversely. The idea of statistical convergence of double sequences has been introduced by Moricz (2003), Mursaleen and Edely (2003) and further studied by Mohiuddine et al. (2012a, b and d;, Mursaleen and Mohiuddine (2009). Let where it is assumed that the series on the right converges for each N ∈ n . We say that a sequence x is A-summable to the limit  if  → n y as .

∞ → 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 a matrix to be regular are known as Silverman-Toeplitz conditions (MADDOX, 1970 In Edely and  have given the notion of statistical A-summability for single sequences and statistical A-summability for double sequences has recently been studied in (BELEN et al., 2012). Let We say that x is statistically A-summable to L if for Note that if a sequence is bounded and A-statistically convergent to L, then it is A-summable to L; hence it is statistically A-summable to L but not conversely [see Edely and , the Cesàro matrix, then statistical A-summability is reduced to statistical summability (C, 1) due to Moricz (MORICZ, 2002). We say that a sequence In the following example we exhibit that a sequence is statistically summable (C, 1) but not statistically convergent. Define a sequence Let It is to be noted that any function continuous and bounded on I.
Theorem 1: Let ) ( n L be a sequence of positive linear operators from In this paper, we use the notion of statistical summability (C, 1) to prove a Korovkin type approximation theorem for functions of two variables with the help of test functions

Erkus and Duman (2005) have given the
the space of all bounded and continuous real valued functions on K equipped with norm It is to be noted that any function ) (K H f * ∈ ω is bounded and continuous on K, and a necessary and sufficient condition for We prove the following result: if and only if Proof.: Since each of the functions Combining (10) and (11), we get (12) After using the properties of f, a simple calculation gives that where: Then from (13), we see that Hence conditions (6)-(9) imply the condition (5).
This completes the proof of the theorem. For Then for all if and only if

Statistical rate of convergence
In this section, using the concept of statistically summable (C, 1) we study the rate of convergence of positive linear operators with the help of the modulus of continuity. Let us recall, for ∈ f ) (K H * ω where: We have the following result: Theorem 3: Let ) ( k T be a sequence of positive linear operators from where: Then for all   (19) and (20) we conclude This completes the proof of the theorem.

Example and the concluding remark
We show that the following double sequence of positive linear operators satisfies the conditions of Theorem 2 but does not satisfy the conditions of Corollary 2 and Theorem 2 of (ERKUS; DUMAN, 2005 ( y x f B u y x f L n n n + It is easy to see that the sequence ) ( n L satisfies the conditions (6), (7), (8) and (9). Hence by Theorem 2, we have On the other hand, the sequence ) ( n L does not satisfy the conditions of Corollary 2 and Theorem 2 of (EDELY et al., 2010), since ) ( n u as well as ) ( n L is neither convergent nor statistically (nor A-statistically) convergent. That is, Corollary 2 and Theorem 2 of (ERKUS; DUMAN, 2005) do not work for our operators . n L Hence our Theorem 2 is stronger than Corollary 2 and Theorem 2 of (ERKUS; DUMAN, 2005).

Conclusion
Korovkin type approximation theorems have recently been proved for different types of summability methods, e.g. staitistcal convergence, A-statistical convergence, Statistical A-summability etc.
In this paper, we have proved such approximation theorem for functions of two variables with the help of test functions x + + + + + by using the notion of statistical summability (C, 1). Through an example, we have also justified that our result was stronger than those of previously proved for ordinary convergence and statistical convergence.