Approximation Theorems for Functions of Two Variables via 𝜎-Convergence

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.