A Note on the Statistical Approximation Properties of the Modified Discrete 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 that
and δ(E) ≤ δ(E1) + δ(E2)
because of
and
.
The proof is completed.