ABSTRACT

In this present paper, firstly, the modified positive operators and its discrete operators are constructed. Then, we investigate the statistical approximation properties and rates of convergence by using modulus of continuity of these positive linear operators. Finally, we obtain the rate of statistical convergence of truncated operators.

1. Introduction

First of all, let us recall the concept of statistical convergence. The natural density (or density) of the set is denoted as.

For

whenever the limit exists (see e.g. [1]) if for every

then we say that a sequence is said to be statistcally convergent to a number of L (see Fast in [2]).

The concept of statistical convergence is very important in approximation theory because although any sequence which is convergent in ordinary sense is statistically convergent, but contrary can not be true all the time. For instance;

If we choose as

then we can easily say that it is statistically convergent to but not convergent in ordinary sense when.Recently, linear positive operators and their Korovkin type statistical approximation properties have been investigated by many authors. It is well-known that lots of operators were defined with infinite series. Details can be found in [3]. For example, n-th Favard-Mirakjan-Szász operator was defined by

for every f belonging to Banach lattice, and where

is endowed with the norm.

In [4], Doğru investigate the weighted approximation properties of general positive linear operators on infinite intervals. Later, in 2002, weighted approximation properties of Szász-type operators are investigated by same author in [5]. In this note, we investigate the statistical approximation properties considering only the partial sums of the operators. In [6], J. Grof studied on the operator

(1)

where and he verified that if is a sequence of positive integers such that

then for all

and Here, f satisfies the inequality

In 1984, Heintz-Gerd Lehnhoff [7] studied the following Modified Szász operators

(2)

where,

Grof and Lehnhoff obtained the conditions which ensure the convergence of the operators to f.

Notice that the notation shows the largest integer and it is not exceeding the number.

The main aim of this paper is to investigate the statistical approximation properties of the operators which constructed and examined the ordinary approximation properties by Agratini in [8].

2. Statistical Approximation Properties

Let us recall the operators which were defined by Agratini in [8].

Throughout the paper, K indicates a compact subinterval of and, the j-th monomial,

Let us assume that the following cases for each

1) For every, a sequence of exists such that

a net on is fixed.

2) There is a sequence such that

. Where, is the space of all realvalued functions continuously differentiable in IR⁺.

For this sequence the following conditions

(3)

hold.

3) A positive function , exists with the property,

(4)

By using these requirements the operators were defined as

(5)

where F stands for the domain of containing the set of all continuous functions on for which the series in (5) is convergent.

We note that, with specific choosing these operators turn into the operators mentioned in [1].

Lemma A. [8] Let, , be defined by (5) and be the r-th central moment of. For every , we have the following identities,

(6)

(7)

(8)

A Korovkin type statistical approximation theorem for any sequence of positive linear operators was proved by Gadjiev and Orhan in [9]. First, let us recall this theorem.

Where denotes all functions f that are continuous in [a,b] and bounded all positive axis.

Theorem A. [9] If the sequence of positive linear operators satisfies the conditions

then for any function we have,

Now, we can give the following theorem which includes the satatistical convergence of the operators in (5).

Theorem 1. Let, be the operators defined in (5). If uniformly on K then for every we have,

(9)

Proof. Because of (3) we can easily say that

(10)

and

(11)

We know from (8) that

. By using the linearity of the operator

From (3)

Hence,

In view of we have

(12)

Now, we are able to say in the light of Theorem A that

which ends the proof.

By using modulus of continuity, we mention about the rate of statistical convergence of these operators. First, let us remember the definition of modulus of continuity. Let the modulus of continuity of f, is defined as

Let be defined by (5), for every and We know from Theorem 1 in [8] that

(13)

If we take norm on K and choose, we get Due to

, we have the rates of statistical convergence of the operators in (5).

3. Modified Discrete Operators

In this section, we recall the modified discrete operators which were defined by Agratini in [8] and investigate the statistical approximation properties of these operators. If we specialize the net and function respectively,

(14)

under these assumptions, the requirement of Theorem 1 is fulfilled. Starting from (5) under the additional assumptions (14) Agratini defined,

(15)

where, is a sequence of positive numbers.

The study of these operators were developed in polynomial weighted spaces connected to the weights

For every

the spaces

endowed with the norm

Lemma B. [8] Let be defined by (5) and the assumptions (14) are fulfilled. If, then the central moment of 2m-th order verifies

(16)

Where is a constant depending only on m and the compact K.

Theorem 2. Let be defined by (15). If

holds for every.

Proof. We use the following,

(17)

and for which are the positive constants hold true. Hence,

From this inequality,

(18)

If then On the grounds of and are positive we can write that

(19)

The remaining term is and taking into consideration both (18) and (19)

By using (16)

If we take norm on K we have the following.

By considering the concept of statistical convergence let us define the sets,

It is obvious thatand δ(E) ≤ δ(E₁) + δ(E₂)

because ofand.

The proof is completed.

REFERENCES