Abstract

Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of any sequence of positive linear operators is discussed on weighted spaces, for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of -summability which is a stronger convergence method than ordinary convergence.

1. Introduction

The fundamental theorem of Korovkin [1] on approximation of continuous functions on a compact interval gives conditions in order to decide whether a sequence of positive linear operators converges to identity operator. This theorem has been extended in several directions. One of the most important papers on these extensions is [2] that where the author obtained Korovkin-type theorem on unbounded sets for the weighted continuous functions on semireal axis. Korovkin-type theorems were also studied on -spaces (see [3, 4]).

The extension of Korovkin’s theorem from compact intervals to unbounded intervals for functions that belong to -spaces was obtained by Gadjiev and Aral [5]. We recall some notations presented in that paper. Let denote the set of real numbers. The function is called a weight function if it is positive continuous function on the whole real axis and, for a fixed , satisfying the condition

Let () denote the linear space of measurable, -absolutely integrable functions on with respect to weight function ; that is,

The analogues of (1) and (2) in multidimensional space are given as follows. Let be a positive continuous function in , satisfying the condition and for one has

The authors obtained Korovkin-type theorems for the functions in and also in . The aforementioned results are the extensions of Korovkin’s theorem on unbounded sets and more general functions spaces by ordinary convergence.

On the other hand, most of the classical operators tend to converge to the value of the function being approximated. At the points of discontinuity, they often converge to the average of the left and right limits of the function. However, there are exceptions which do not converge at points of discontinuity (see [6]). In this case matrix summability methods of Cesáro type are strong enough to correct the lack of convergence [7].

Let be a sequence of infinite matrices with non-negative real entries. For a sequence , the double sequence defined by is called -transform of whenever the series that converges for all , , and is said to be -summable to if uniformly in ([8, 9]). If for some matrices , then -summability is the ordinary matrix summability by , and if , for (), and otherwise, then -summability reduces to almost convergence [10]. Replacing the ordinary convergence by -summability some approximation results have been studied in [11–13] and in the special cases [14, 15]. Also, Korovkin-type theorems in weighted space via -summability have been studied in [16, 17].

Our purpose in the present paper is to obtain Korovkin-type theorems on weighted spaces in univariate and multivariate case via -summation process. More precisely, a sequence of positive linear operators from into is called an -summation process on if is -summable to for every ; that is, where it is assumed that the series in (8) converges for each , , and . Considering this fact we extend (8) to space of sequences of linear positive operators to approximate the functions that belong to spaces via matrix summability method.

2. Main Result

Throughout this section we will use the following notations: is the double sequence: and minimum and maximum values of the weight function on finite intervals will be denoted by and , respectively.

Now we present the following main result.

Theorem 1. Let be a sequence of infinite matrices with nonnegative real entries and let be a sequence of positive linear operators from into . Assume that
If then for any function , one has

Proof. Let be the characteristic function of the interval and for any . We can choose a sufficient large such that for every
Using the assumption of the convergence of the series (9) for each , , and and the linearity of the operators , we get
By condition (10), there exists a constant such that
Hence, from (13), we compute
For every function the inequality implies that . Since the space of continuous functions is dense in , given , for each , there exists a continuous function on satisfying the condition for such that
Using the inequalities (15) and (18), we get
On the other hand, since for some , we get the inequality
Now, by supposing that , we get
Since , we can choose the number such that
Using this inequality, we have
As a corollary, we get the following inequality for :
Since is a continuous function on , for any given , there exists a such that
Furthermore, this inequality also holds in the case that and that since and are continuous. So, we have
Using (24) and (26), we can write
Then we obtain the following equality for (14): where . By the hypothesis of theorem and arbitrarity of and as which is desired result.

Now we give an example of a sequence of positive linear operators which satisfies the conditions of Theorem 1 in weighted space .

Example 2. We choose . Note that this selection of satisfies condition (1). Also note that for
Also for each where is the Cesáro matrix; that is,
The Kantorovich variant of the Szász-Mirakyan operators [18] by replacing with an integral mean of over the interval is as follows: where is a sequence of positive real numbers satisfying the condition
It is known that
Furthermore by simple calculations, we obtain
Also,
Hence, conditions (10), (11) are provided which means that for any function , we have

Also, analogue of Theorem 1 for the space of function of several variables can be obtained. Now, we establish this theorem. For the sake of convenient notation, we present our second results on , , instead of to avoid any confusion about the indices of .

Theorem 3. Let be a sequence of infinite matrices with nonnegative real entries and let be a sequence of positive linear operators from into . Assume that
If then for any function , one has

Proof. Considering the characteristic function of the ball and , it is possible to choose a sufficient large such that
On the other hand, by condition (37) there exists a positive constant such that , and so, for given , there exists a continuous function on satisfying the condition for such that
Keeping in mind the fact that the series (9) is a convergence for each , , and , and using the linearity of the operators , which means the linearity of , we get
Let , so we also have where . Furthermore, we can choose such that , and for sufficiently large , we estimate . Using these estimations in (42), we obtain
Since
we can write
Using the conditions of theorem, we have which means that