Kamil Demirci

Statistical Convergence of Double Sequences on Probabilistic Normed Spaces

The concept of statistical convergence was presented by Steinhaus in 1951. This concept was extended to the double sequences by Mursaleen and Edely in 2003. Karakus has recently introduced the concept of statistical convergence of ordinary (single) sequence on probabilistic normed spaces. In this paper, we define statistical analogues of convergence and Cauchy for double sequences on probabilistic normed spaces. Then we display an exampl e such that our method of convergence is stronger than usual convergence on probabilistic normed spaces. Also we give a useful characterization for statistically convergent double sequences.

Four-dimensional matrix transformation and rate of A-statistical convergence of periodic functions

In this paper, using the concept of A-statistical convergence for double real sequences, we obtain a Korovkin type-approximation theorem for double sequences of positive linear operators defined on the space of all 2@p-periodic and real valued continuous functions on the real two-dimensional space. Furthermore, we display an application which shows that our new result is stronger than its classical version. Also, we study rates of A-statistical convergence of a double sequence of positive linear operators acting on this space. Finally, displaying an example, it is shown that our statistical rates are more efficient than the classical aspects in the approximation theory.

Statistical A-summability of positive linear operators

In this paper, using the concept of statistical A-summability which is stronger than the A-statistical convergence we provide a Korovkin-type approximation theorem. We also compute the rates of statistical A-summability of a sequence of positive linear operators.

A KOROVKIN TYPE APPROXIMATION THEOREM FOR DOUBLE SEQUENCES OF POSITIVE LINEAR OPERATORS OF TWO VARIABLES IN A-STATISTICAL SENSE

In this paper, we obtain a Korovkin type approximation the- orem for double sequences of positive linear operators of two variables from Hw (K) to C (K) via A-statistical convergence. Also, we construct an example such that our new approximation result works but its classi- cal case does not work. Furthermore, we study the rates of A-statistical convergence by means of the modulus of continuity.

Four-dimensional matrix transformation and the rate of A-statistical convergence of continuous functions

Motivated by our earlier work on the statistical approximation of continuous functions by positive linear operators of two variables, we study rates of A-statistical convergence of a sequence of positive linear operators acting on the space of all continuous real valued functions on any D compact subset of the real two-dimensional space. Furthermore, displaying an example, it is shown that our statistical rates are more efficient than the classical aspects in the approximation theory.

Statistical σ approximation to max-product operators

In this paper, using the concept of statistical @s-convergence which is stronger than statistical convergence, we obtain a statistical @s approximation theorem for a general sequence of max-product operators, including Shepard type operators, although its classical limit fails. We also compute the corresponding statistical @s rates of the approximation.

Korovkin-Type Theorems for Modular --Statistical Convergence

We deal with a new type of statistical convergence for double sequences, called --statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

Approximation in statistical sense to n-variate B-continuous functions by positive linear operators

Our primary interest in the present paper is to prove a Korovkintype approximation theorem for sequences of positive linear operators defined on the space of all real valued n-variate B-continuous functions on a compact subset of the real n-dimensional space via statistical convergence. Also, we display an example such that our method of convergence is stronger than the usual convergence.

Equi-statistical σ -convergence of positive linear operators

In this paper we introduce the notion of equi-statistical @s-convergence which is stronger than the uniform convergence (in the ordinary sense), statistical uniform convergence and statistical uniform @s-convergence. Then, we also give its use in the Korovkin-type approximation theory. We also compute the rate of equi-statistical @s-convergence of a sequence of positive linear operators.

Statistical approximation to Bögel-type continuous and periodic functions

In this paper, considering A-statistical convergence instead of Pringsheim’s sense for double sequences, we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued Bögel-type continuous and periodic functions on the whole real two-dimensional space. A strong application is also presented. Furthermore, we obtain some rates of A-statistical convergence in our approximation.