
The concept of
The idea of statistical convergence was given by Zygmund [36] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was formally introduced by Steinhaus [35] and Fast [16] and later reintroduced by Schoenberg [34]. Various generalizations and applications of statistical convergence have been studied by
Statistical convergence depends on the natural density of subsets of the set ℕ = {1, 2, 3, …}. The natural density
where |{
The concept of statistical convergence was studied by Kolk [22] in the more general setting of normed spaces.
Let
and we write it as
Recall [25, 32] that a modulus
If
Aizpuru
For any unbounded modulus
in case this limit exists.
Let
and we write it as
For any unbounded modulus
The theory of approximation is an area of mathematical analysis, which, at its core, is concerned with the approximation of functions by simpler and more easily calculated functions. In the fifties, the theory of approximation of functions by positive linear operators developed a lot, when Popoviciu [31], Bohman [11] and Korvokin [23, 24], discovered, independently, a simple and easily applicable criterion to check if a sequence of positive linear operators converges uniformly to the function to be approximated. This criterion says that the necessary and sufficient condition for the uniform convergence of the sequence (
Due to this classical result, the monomials
Statistical convergence had not been examined in approximation theory until 2002. The Korovkin first and second approximation theorems were first proved via statistical convergence by Gadjiev and Orhan [18] and Duman [14], in years 2002 and 2003, respectively. After this, Korovkin-type approximation theorems have been studied via various summability methods by many mathematicians [4, 13, 21, 26, 27, 28]. Quite recently Bhardwaj and Dhawan [8] have obtained
In 1970, Boyanov and Veselinov [12] have proved the Korovkin theorem on
Before proceeding to establish the proposed results, we recall [20, 30] that for any linear spaces
the mapping
if
in order to highlight the argument of the function
if L is a positive linear operator, then for every
We begin this section by recalling the classical Korovkin type approximation theorem due to of Boyanov and Veselinov [12].
The statistical analog of this theorem, given by Duman et al. [15], is as follows.
We now state and prove an
Since each of 1,
It is easy to prove that for a given
Putting
This means,
In fact, if |
and (
In view of monotonicity and linearity of the operators
Note that
But
Using (
Now, let us estimate
Using (
Therefore, using the fact that |
where
Now, write
It is easy to see that
which yields that
and using (
Since every convergent sequence is
Our next example shows that there may exist a sequence of positive linear operators which satisfies the conditions of Theorem 2.3 but does not satisfy the conditions of Theorem 2.1, thereby showing that our result is stronger than the classical one.
Consider the sequence
where (
and (
Now making use of the fact (Remark 1.4 ) that every convergent sequence is
On the other hand,
and so,
from where it follows that (
Since every
Our next example shows that there may exist a sequence of positive linear operators which satisfies the conditions of Theorem 2.2 but not of Theorem 2.3, thereby showing that the statistical analog is stronger than the
Consider the sequence
where (
From Remarks 2.4, 2.6 and Examples 2.5, 2.7 it follows that the classical Korovkin type approximation theorem of Boyanov and Veselinov implies its
In this section, using
Let
In this case, we write
(
(
(i) Assume that
Now by taking the limit as
where
Recall [20] that for a continuous function
is called its usual modulus of continuity.
To estimate the rate of convergence of a sequence of positive linear operators defined from
defined for every δ ≥ 0 and every function
where
It is well known ( e.g. see [20]) that for any δ > 0
We are now in a position to give the promised estimation.
Let
Putting
where
Now, for a given
Then it follows from (
Now, since
from where it follows that
The authors wish to thank the referee for his valuable suggestions, which have improved the presentation of the paper.