On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables

Kadak, Uğur

doi:10.1016/j.jmaa.2016.05.062

Cited by 45 publications

(14 citation statements)

References 25 publications

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“…In the year of 2016, Kadak [25] introduced weighted statistical convergence involving (p, q)-integers and proved some approximation theorems for functions of two variables. Subsequently, it was extended to the generalized difference sequences involving (p, q)-gamma function and accordingly associated approximation theorems were proved (see [24]).…”

Section: Remark 13mentioning

confidence: 99%

A certain class of deferred weighted statistical B-summability involving (p,q)-integers and analogous approximation theorems

Zraiqat

Paikray

Dutta

2019

Filomat

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The preliminary idea of statistical weighted B-summability was introduced by Kadak et al. [27]. Subsequently, deferred weighted statistical B-summability has recently been studied by Pradhan et al. [38]. In this paper, we study statistical versions of deferred weighted B-summability as well as deferred weighted B-convergence with respect to the difference sequence of order r (> 0) involving (p, q)-integers and accordingly established an inclusion between them. Moreover, based upon our proposed methods, we prove an approximation theorem (Korovkin-type) for functions of two variables defined on a Banach space C B (D) and demonstrated that, our theorem effectively improves and generalizes most (if not all) of the existing results depending on the choice of (p, q)-integers. Finally, with the help of the modulus of continuity we estimate the rate of convergence for our proposed methods. Also, an illustrative example is provided here by generalized (p, q)-analogue of Bernstein operators of two variables to demonstrate that our theorem is stronger than its traditional and statistical versions. 1. Introduction, Preliminaries and Motivation Let ω be the set of all real valued sequences and call any subspace of ω the sequence space. Let (x k) be a sequence with real or complex terms. Suppose ∞ is the class of all bounded linear sequence spaces and let c, c 0 be the respective classes for convergent and null sequences with real or complex terms. We have, x ∞ = sup k |x k | (k ∈ N), and we recall here that under this norm, the above mentioned spaces are all Banach spaces. The space of difference sequence was initially studied by Kızmaz [30] and then it was extended to the difference sequence of natural order r (r ∈ N 0 := {0} ∪ N) by defining λ(∆ r) = {x = (x k) : ∆ r (x) ∈ λ, λ ∈ (∞ , c 0 , c)} ;

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Section: Remark 13mentioning

confidence: 99%

A certain class of deferred weighted statistical B-summability involving (p,q)-integers and analogous approximation theorems

Zraiqat

Paikray

Dutta

2019

Filomat

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“…The (p, q)and q-Dunkl analogues have been studied by several authors (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). For the most recent work on (p, q)-approximation, we refer to [25][26][27].…”

Section: Introductionmentioning

confidence: 99%

A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus

2019

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In this paper, we define the ( p , q ) -variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. The extra parameter p provides more flexibility in approximation and plays an important role in symmetrizing these newly-defined operators.

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“…Recently, statistical convergence has been a dynamic research area due to the fact that it is more general than classical convergence and such theory is discussed in the study of Fourier Analysis, Number Theory and Approximation Theory. For more details, see [2], [5], [7], [10], [12], [14] and [15].…”

Section: Introductionmentioning

confidence: 99%

Statistical deferred Cesàro summability and its applications to approximation theorems

Jena

Kumar

Misra

2018

Filomat

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Statistical (C, 1) summability and a Korovkin type approximation theorem has been proved by Mohiuddine et al. [20] (see [S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical summability (C, 1) and a Korovkin type approximation theorem, J. Inequal. Appl. 2012 (2012), Article ID 172, 1-8). In this paper, we apply statistical deferred Cesàro summability method to prove a Korovkin type approximation theorem for the set of functions 1, e −x and e −2x defined on a Banach space C[0, ∞) and demonstrate that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also establish a result for the rate of statistical deferred Cesàro summability method. Some interesting examples are also discussed here in support of our definitions and results.

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On weighted statistical convergence based on (p,q)-integers and related approximation theorems for functions of two variables

Cited by 45 publications

References 25 publications

A certain class of deferred weighted statistical B-summability involving (p,q)-integers and analogous approximation theorems

A certain class of deferred weighted statistical B-summability involving (p,q)-integers and analogous approximation theorems

A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus

Statistical deferred Cesàro summability and its applications to approximation theorems

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