Journal of Function Spaces

Volume 2014 (2014), Article ID 350474, 5 pages

http://dx.doi.org/10.1155/2014/350474

Research Article

Approximation Theorems for Functions of Two Variables via -Convergence

Mohammed A. Alghamdi

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 23 October 2013; Accepted 13 December 2013; Published 10 February 2014

Academic Editor: M. Mursaleen

Copyright © 2014 Mohammed A. Alghamdi. 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.

Abstract.

Çakan et al. (2006) introduced the concept of -convergence for double sequences. In this work, we use this notion to prove the Korovkin-type approximation theorem for functions of two variables by using the test functions 1, , , and and construct an example by considering the Bernstein polynomials of two variables in support of our main result.

1. Introduction and Preliminaries

In [1], Pringsheim introduced the following concept of convergence for double sequences. A double sequence is said to be to the number in Pringsheim’s sense (shortly, to ) if for every there exists an integer such that whenever . In this case is called the of .

A double sequence of real or complex numbers is said to be if . We denote the space of all bounded double sequences by .

If and is -convergent to , then is said to be   to (shortly, to ). In this case is called the of the double sequences . The assumption of being   -convergent was made because a double sequence which is -convergent is not necessarily bounded.

Assume that is a one-to-one mapping from the set (the set of natural numbers) into itself. A continuous linear functional on the space of bounded single sequences is said to be an or a if and only if (i) when the sequence has for all , (ii) , where , and (iii) for all .

Throughout this paper we consider the mapping which has no finite orbits; that is, for all integer and , where denotes the iterate of at . Note that a -mean extends the limit functional on the space of convergent single sequences in the sense that for all (see [2]). Consequently, the set of bounded sequences all of whose -means are equal. We say that a sequence is - if and only if . Schaefer [3] defined and characterized the -conservative, -regular, and -coercive matrices for single sequences by using the notion of -convergence. If , then the set is reduced to the set of almost convergent sequences due to Lorentz [4].

In 2006, Çakan et al. [5] presented the following definition of -convergence for double sequences and established core theorem for -convergence and later on this notion was studied by Mursaleen and Mohiuddine [6–8]. A double sequence of real numbers is said to be - to a number if and only if , where

while here the limit means -limit. Let us denote by the space of -convergent double sequences . If is translation mapping, then the set is reduced to the set of almost convergent double sequences [9]. Note that .

Example 1. Let be a double sequence such that

for all . Then is -convergent to zero (for ) but not convergent.

Suppose that is the space of all functions continuous on . It is well known that is a Banach space with the norm

The classical Korovkin approximation theorem is given as follows (see [10, 11]).

Theorem 2. Let be a sequence of positive linear operators from into and , for , where , , and . Then , for all .

In [12], Mohiuddine obtained the Korovkin-type approximation theorem through the notion of almost convergence for single sequences and proved some interesting results. Such type of approximation theorems for the function of two variables is proved in [13, 14] through the concept of almost convergence and statistical convergence of double sequences, respectively. Recently, Mohiuddine et al. [15] determined the Korovkin-type approximation theorem by using the test functions , , and through the notion of statistical summability . Quite recently, by using the concept of -statistical convergence, Mohiuddine and Alotaibi [16] proved the Korovkin-type approximation theorem for functions of two variables. For more details and some recent work on this topic, we refer to [17–21] and references therein. In this work, we apply the notion of -convergence to prove the Korovkin-type approximation theorem by using the test functions , , , and . We apply the classical Bernstein polynomials of two variables to construct an example in support of our result.

2. Main Result

Now, we prove the classical Korovkin-type approximation theorem for the functions of two variables through -convergence.

By , we denote the set of all two dimensional continuous functions on , where . Let be a linear operator from into . Then, a linear operator is said to be positive provided that implies .

Theorem 3. Suppose that is a double sequence of positive linear operators from into and satisfying the following conditions:

which hold uniformly in . Then for any function bounded on the whole plane, one has

uniformly in .

Proof. Since and is bounded on the whole plane, we have

Therefore,

Also we have that is continuous on ; that is,

From (7) and (8), putting and , we obtain

or

Now, we operate on the above inequality since is monotone and linear. We obtain

Therefore

But

From (12) and (13), we get

Now

Using (14), we obtain

Since is arbitrary, we can write

Similarly,

Thus, we have

Taking the limit and from (4), we obtain

Theorem 4. Suppose a double sequence of positive linear operators on such that

If

where , , , and , then

for any function bounded on the whole plane.

Proof. From Theorem 3, we have that if (22) holds then

which is equivalent to

Now

Therefore

Hence, using the hypothesis, we get

That is, (23) holds.

3. Example and the Concluding Remark

In this section, we prove that our theorem is stronger than Theorem 2. Let us consider the following Bernstein polynomials (see [22]) of two variables:

Let be defined by

where the double sequence is defined by (2) in Section 1. Then

Also, satisfies (4). Hence, we have

Since , we have . Thus

But Theorem 3 does not hold, since the limit does not exist as . Therefore we conclude that our Theorem 3 is stronger than the classical Korovkin theorem for functions of two variables due to Volkov [23].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

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