We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability . We also study the rate of statistical summability 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 [1] and Steinhaus [2] independently introduced an extension of the usual concept of sequential limit which is called statistical convergence.

The number sequenceis said to be statistically convergent to the number provided that for each, wheredenotes the number of elements ofnot exceeding. In this case we write .

The notion of statistical convergence of double sequenceshas been introduced and studied in [3, 4] independently in the same year, 2003.

Letbe a two-dimensional set of positive integers and letbe the numbers ofinsuch thatandThen the two-dimensional analogue of natural density can be defined as follows.

The lower asymptotic density of a set is defined as In this case the sequencehas a limit in Pringsheim’s sense then we say thathas a double natural density and is defined as

A real double sequenceis said to be statistically convergent to the number if for each, the set has double natural density zero. In this case we write.

Ifis statistically convergent, thenneed not be convergent. Also it is not necessarily bounded. For example, letbe defined as It is easy to see that, since the cardinality of the setfor every. Butis neither convergent nor bounded.

Móricz [5] introduced the notion of statistical summabilityA double sequenceis said to be statistically summableto the numberif for every, where is themean ofThus, the double sequenceis statistically summableto if and only if the sequenceis statistically convergent to . In this case we write . Note that if a double sequence is bounded then implies.

Korovkin type approximation theorems (cf. [610]) are useful tools to check whether a given sequenceof positive linear operators onof all continuous functions on the real intervalis 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 , , and in the spaceas well as for test functions , , and in the space of all continuous -periodic functions on the real line.

We know thatis a Banach space with norm

We denote bythe space of all-periodic functionswhich is a Banach space with

After the paper of Gadjiev and Orhan [11], 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-summability of double sequences; in [1316], the authors have used, respectively, statistical convergence and-statistical convergence of double sequences; and in [17, 18], the authors used almost summability. For some more related work, we refer to [1922].

In this paper, we present the Korovkin type approximation theorem for periodic functions via statistical summabilityand also study the rate of statistical summabilityof a double sequence of positive linear operators defined fromintowhereis the space of all-periodic and real valued continuous functions onequipped with the norm

2. Main Result

First, we state the result due to [15] for-statistical convergence of double sequences.

Theorem 1. Letbe a double sequence of positive linear operators acting fromintoThen, for all if and only if where,???, and.

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

Corollary 2. Letbe a double sequence of positive linear operators acting fromintoThen, for all if and only if We prove the following result.

Theorem 3. Letbe a double sequence of positive linear operators acting fromintoThen, for all if and only if

Proof. Since each of the functions , , , , and belongs to , necessity follows immediately from (15). Let condition (16) hold andLetandbe closed subintervals each of length of . . By the continuity ofat, it follows that for given there is a numbersuch that for all wheneverSinceis bounded, it follows that for all.
For all, it is well known that whereSince the functionis-periodic, the inequality (19) holds for. Then, we obtain where. Now, taking, we get Now for a givenchoosesuch that. Define the following sets: where. Then by (21) and so
Now using (16), we get

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 where Observe that Define a double sequenceby, .
We observe that is neither-convergent nor statistically convergent but
Let us define the operatorsby Then, observe that the double sequence of positive linear operatorsdefined by (30) satisfies all hypotheses of Theorem 3. Hence, by (28), we have, for all, Sinceis neither-convergent nor statistically convergent, the sequencegiven by (30) is also neither-convergent nor statistically convergent to the function. So, we conclude that Corollary 2 and Theorem 1 do not work for the operatorsgiven by (30) while Theorem 3 still works. Hence, we conclude that-version is stronger than that of-version as well as statistical version.

3. Rate of Statistical Summability

Letbe a positive nonincreasing double sequence. We say that a double sequenceis statistically summableto the numberwith the rateif for every, In this case, we writeas.

Now, we recall the notion of modulus of continuity. The modulus of continuity of , denoted by for , is defined by It is well known that Then we have the following result.

Theorem 5. Letbe a double sequence of positive linear operators acting from into . Let and be two positive non-increasing sequences. Suppose that(i),(ii), whereand Then, for all, where.

Proof. Let and . Let; we have the following cases.
Case I. If,, thenand. Therefore by (34), we have
Case II. If. Letbe an integer such that; then Similarly, in the other two cases when, and , , we obtain (37).
Now, using the definition of modulus of continuity and the linearity and the positivity of the operatorswe get Taking supremum overon both sides of the above inequality and let We obtain where. Let . Now for a givendefine the following sets: Then. Further define We see that. Therefore. Therefore, since, we conclude that for every Using conditions (i) and (ii), we get.


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