Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems

Srivástava, H. M.; Jena, Bidu Bhusan; Misra, U. K.

doi:10.1007/s13398-017-0442-3

Cited by 48 publications

(47 citation statements)

References 16 publications

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“…In this section, we extend the result of Srivastava et al [24] by using the notion of statistically deferred Nörlund summability of a real sequence over a Banach space.…”

Section: A New Korovkin-type Approximation Theoremmentioning

confidence: 80%

“…We now present the following example for the sequence of positive linear operators that does not satisfy the associated conditions of the Korovkin approximation theorems proved previously in [24,33], but it satisfies the conditions of our Theorem 2. Consequently, our Theorem 2 is stronger than the earlier findings of both Srivastava et al [24] and Paikray et al [33].…”

Section: Theorem 2 Letmentioning

confidence: 90%

“…Definition 1. (see [24]) Let (a n ) and (b n ) be sequences of non-negative integers and let (p n ) and (q n ) be the sequences of non-negative real numbers. A real sequence {x n } n∈N is deferred Nörlund statistically convergent to if, for every > 0, {m :…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…Subsequently, Karakaya and Chishti [21] introduced and studied the basic idea of the weighted statistical convergence and it was then modified by Mursaleen et al [22]. Furthermore, Srivastava et al [23,24], studied the notion of the deferred weighted as well as deferred Nörlund statistical convergence and used these notions to prove certain Korovkin-type approximation theorem with some new settings. Later on, some fundamental concept of the deferred Cesàro statistical convergence as well as the statistical deferred Cesàro summability, together with the associated approximation theorems was introduced by Jena et al [25].…”

Section: Introductionmentioning

confidence: 99%

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Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

2020

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The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions.

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“…In this section, we extend the result of Srivastava et al [24] by using the notion of statistically deferred Nörlund summability of a real sequence over a Banach space.…”

Section: A New Korovkin-type Approximation Theoremmentioning

confidence: 80%

Section: Theorem 2 Letmentioning

confidence: 90%

Section: Preliminaries and Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

2020

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“…Furthermore, Kadak et al [27] presented the idea of statistical weighted B-summability and established some approximation theorems on that basis. Very recently, Srivastava et al [41] introduced the deferred weighted (Nörlund) summability of a sequence and accordingly proved Korvokin-type approximation theorems based on equi-statistical convergence.…”

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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Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

Srivástava

Jena

2021

Math Methods in App Sciences

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In this paper, we investigate and study the concepts of statistical product convergence and statistical product summability via deferred Cesàro and deferred weighted product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two nice and potentially useful concepts. Also, based upon our proposed concepts, we state and prove a set of new Korovkin‐type approximation theorems for a martingale sequence over a Banach space. Moreover, we demonstrate that our approximation theorems effectively extend and improve most (if not all) of the previously existing results (both in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are quite stronger than the traditional and statistical versions of different theorems existing in the literature. We also suggest a direction for future researches on this subject, which are based upon the basic (or q‐) calculus, but not upon the trivial and inconsequential variations involving the so‐called (p, q)‐calculus.

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Generalized equi-statistical convergence of the deferred Nörlund summability and its applications to associated approximation theorems

Cited by 48 publications

References 16 publications

Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem

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

Statistical product convergence of martingale sequences and its applications to Korovkin‐type approximation theorems

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