Statistical A-summability of double sequences and a Korovkin type approximation theorem (original) (raw)

Statistical (C,1) (E,1) summability and Korovkin’s theorem

Filomat, 2016

Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. This approximation theorem was extended to more general space of sequences via different way such as statistical convergence, summation processes. In this work, we introduce a new type of statistical product summability, that is, statistical (C, 1) (E, 1) summability and further apply our new product summability method to prove Korovkin type theorem. Furthermore, we present a rate of convergence which is uniform in Korovkin type theorem by statistical (C, 1) (E, 1) summability.

Korovkin Second Theorem via -Statistical -Summability

Abstract and Applied Analysis, 2013

Korovkin type approximation theorems are useful tools to check whether a given sequence of positive linear operators on of all continuous functions on the real interval is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, , and in the space as well as for the functions 1, cos, and sin in the space of all continuous 2-periodic functions on the real line. In this paper, we use the notion of -statistical -summability to prove the Korovkin second approximation theorem. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from into .

Statistical summability (C,1)(C, 1) (C,1) and a Korovkin type approximation theorem

Journal of Inequalities and Applications, 2012

Abstract. The concept of statistical summability (C, 1) has recently been introduced by Moricz [Jour. Math. Anal. Appl., 275 (2002) 277–287]. In this paper, we use this notion of summability to prove the Korovkin type approximation theorem by using the test functions 1, e− x, e− 2x. We also give here the rate of statistical summability (C, 1) and apply the classical Baskakov operator to construct an example in support of our main result. Keywords and phrases: Statistical convergence; Statistical summability (C, 1); Positive linear ...

Korovkin second theorem via B− statistical A− summability

Abstract. Korovkin type approximation theorems are useful tools to check whether a given sequence (Ln) n≥ 1 of positive linear operators on C [0, 1] of all continuous functions on the real interval [0, 1] is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them.

F-Relative A-Summation Process for Double Sequences and Abstract Korovkin Type Theorems

Hacettepe Journal of Mathematics and Statistics, 2021

In this paper, we first introduce the notions of F-relative modular convergence and F-relative strong convergence for double sequences of functions. Then we prove some Korovkin-type approximation theorems via F-relative A-summation Process on modular spaces for double sequences of positive linear operators. Also, we present a non-trivial application such that our Korovkin-type approximation results in modular spaces are stronger than the classical ones and we present some estimates of rates of convergence for abstract Korovkin-type theorems. Furthermore, we relax the positivity condition of linear operators in the Korovkin theorems and study an extension to non-positive operators.

On generalized statistical convergence of double sequences via ideals

ANNALI DELL'UNIVERSITA' DI FERRARA, 2012

The concept of statistical convergence is one of the most active area of research in the field of summability. Most of the new summability methods have relation with this popular method. In this paper we generalize the notions of statistical convergence, (λ, μ)-statistical convergence, (V, λ, μ) summability and (C, 1, 1) summability for a double sequence x = (x jk ) via ideals. We also establish the relation between our new methods.

Weighted statistical convergence and its application to Korovkin type approximation theorem

Applied Mathematics and Computation, 2012

The concept of weighted statistical convergence was introduced and studied by Karakaya and Chishti (2009) [7]. In this paper, we modify the definition of weighted statistical convergence and find its relationship with the concept of statistical summability ðN; p n Þ due to Moricz and Orhan (2004) [10]. We apply this new summability method to prove a Korovkin type approximation theorem by using the test functions 1; e Àx ; e À2x. We apply the classical Baskakov operator to construct an example in support of our result.

Statistical αβ-summability and Korovkin type approximation theorem

Filomat

In this study, we define [N?,??]q - summability and statistical (N?,??) summability. We also establish some inclusion relation and some related results for this new summability methods. Further we apply Korovkin type approximation theorem through statistical (N?,??) summability and we apply the classical Bernstein operator to construct an example in support of our result. Furthermore, we present a rate of convergence which is uniform in Korovkin type theorem by statistical (N?,??) summability.

Statistical Convergence of Double Sequences of Order

Journal of Function Spaces and Applications

We intend to make a new approach and introduce the concepts of statistical convergence of order and strongly -Cesàro summability of order for double sequences of complex or real numbers. Also, some relations between the statistical convergence of order and strong -Cesàro summability of order are given.