Abstract and Applied Analysis

Volume 2013 (2013), Article ID 514174, 7 pages

http://dx.doi.org/10.1155/2013/514174

Research Article

A Generalization of Lacunary Equistatistical Convergence of Positive Linear Operators

Yusuf Kaya and Nazmiye Gönül

Department of Mathematics, Faculty of Arts and Sciences, Bulent Ecevit University, 67100 Zonguldak, Turkey

Received 25 January 2013; Revised 17 April 2013; Accepted 18 April 2013

Academic Editor: Adem Kılıçman

Copyright © 2013 Yusuf Kaya and Nazmiye Gönül. 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.

In this paper we consider some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. In particular we study lacunary equi-statistical convergence of approximating operators on spaces, the spaces of all real valued continuous functions de…ned on and satisfying some special conditions.

1. Introduction

Approximation theory has important applications in the theory of polynomial approximation, in various areas of functional analysis, numerical solutions of integral and differential equations [1–6]. In recent years, with the help of the concept of statistical convergence, various statistical approximation results have been proved [7]. In the usual sense, every convergent sequence is statistically convergent, but its converse is not always true. And, statistical convergent sequences do not need to be bounded.

Recently, Aktuğlu and Gezer [8] generalized the idea of statistical convergence to lacunary equi-statistical convergences. In this paper, we first study some Korovkin type approximation theorems via lacunary equi- statistical convergence in spaces. Then using the modulus of continuity, we study rates of lacunary equi-statistically convergence in .

We recall here the concepts of equi-statistical convergence and lacunary equi-statistical convergence.

Let and belong to , which is the space of all continuous real valued functions on a compact subset of the real numbers. is said to be equi-statistically convergent to on and denoted by if for every , the sequence of real valued functions

converges uniformly to the zero function on , which means that

A lacunary sequence is an integer sequence such that

In this paper the intervals determined by will be denoted by , and the ratio will be abbreviated by .

Let be a lacunary sequence then is said to be lacunary equi-statistically convergent to on and denoted by if for every , the sequence of real valued functions defined by

uniformly converges to zero function on , which means that

A Korovkin type approximation theorem by means of lacunary equi-statistical convergence was given in [8]. We can state this theorem now. An operator defined on a linear space of functions is called linear if , for all , and is called positive, if , for all , . Let be a compact subset of , and let be the space of all continuous real valued functions on .

Lemma 1 (see [8]). Let be a lacunary sequence, and let be a sequence of positive linear operators satisfying

then for all ,

We turn to introducing some notation and the basic definitions used in this paper. Throughout this paper . Let

and

Consider the space of all real-valued functions defined on and satisfying

where is the modulus of continuity defined by

(see [9]). Let , then the norm on is given by

and also denote the valued of at a point is denoted by [10, 11].

is the type of modulus of continuity for the functions of two variables satisfying the following properties: for any real numbers , , , , , and ,

(i) and are nonnegative increasing functions on ,(ii), (iii), (iv).

The space is of all real-valued functions defined on and satisfying

It is clear that any function in is continuous and bounded on .

2. Lacunary Equistatistical Approximation

In this section, using the concept of Lacunary equistatistical convergence, we give a Korovkin type result for a sequence of positive linear operators defined on , the space of all continuous real valued functions on the subset of , and the real -dimensional space. We first consider the case of . Following [7] we can state the following theorem.

Theorem 2. Let be a lacunary sequence, and let be a sequence of positive linear operators from into . is satisfying , , where , ,

then for all ,

Proof. Let be a fixed point, , and assume that (14) holds. For every , there exist such that holds for all satisfying

Let

Hence,

where denotes the characteristic function of the set . Observe that

Using (18), (19), and we have

where .

By the linearity and positivity of the operators and by (18), we have

Hence, we get

where . For a given , choose such that . Define the following sets:

where  . Then from (22) we clearly have

Therefore define the following real valued functions:

where . Then by the monotonicity and (24) we get

for all , and this implies the inequality

Taking limit in (27) as and using (14) we have

Then for all , we conclude that

Now replace by and by an induction, we consider the modulus of continuity type function as in the function . Then let be the space of all real-valued functions satisfying

Therefore, using a similar technique in the proof of Lemma 1 one can obtain the following result immediately.

Theorem 3. Let be a lacunary sequence, and let be a sequence of positive linear operators from into . is satisfying

where , ,

Then for all ,

Assume that , . One considers the following positive linear operators defined on :

where , and .

Lemma 4. Let be a lacunary sequence, and let

be a sequence of positive linear operators from into . If is satisfying

then for all ,

Proof. Assume that (36) holds, and let . Since

it is clear that, for all ,

Now, by assumption we have

Using the definition of , we get

Since

we get

So, we have

The fact that and using a similar technique as in the proof of Lemma 1, we get

Hence we have

Also we have

To see this, by the definition of , we first write

Then,

which implies that

Since

we get

Thus . Therefore we obtain that for all , .

3. Rates of Lacunary Equistatistical Convergence

In this section we study the order of lacunary equi-statistical convergence of a sequence of positive linear operators acting on , where . To achieve this we first consider the case of .

Definition 5. A sequence is called lacunary equi-statistically convergent to a function with rate if for every ,

where is given in Lemma 1. In this case it is denoted by

Lemma 6. Let and be two sequences of functions in , with

Then one has

where .

Proof. Assume that , and , on . For all , consider the following functions:

Then we have

and hence

Taking limit as and using the assumption complete the proof.

Now we give the rate of lacunary equi-statistical convergence of a positive linear operators to with the help of modulus of continuity.

Theorem 7. Let , and let be a sequence of positive linear operators. Assume that

(i) on ,(ii) on with

Then

where .

Proof. Let and . Use

where . Using inequality (62), conditions (i) and (ii) we get

so we have

Finally we give the rate of lacunary equi-statistical convergence for the operators by using the Peetre's -functional in the space . The Peetre -functional of function is defined by

where

Theorem 8. Let and be the sequence of Peetre's K-functional. If

on , then

Proof. For each , we get

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