Generalized deferred Cesàro equi-statistical convergence and analogous approximation theorems

Parida, Priyadarsini; Paikray, Susanta Kumar

doi:10.22199/issn.0717-6279-2020-02-0020

Cited by 13 publications

(12 citation statements)

References 21 publications

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“…Moreover, the introduction of statistical probability convergence has enhanced the glory of this development. For some recent research works in this direction, see [2], [3], [4], [6], [7], [9], [11], [16] and [20].…”

Section: Introductionmentioning

confidence: 99%

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

Jena

Paikray

2022

AMI

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In this paper, we investigate and study the notions of statistical product convergence and statistical product summability via deferred Cesàro and deferred Nörlund product means for martingale sequences of random variables. We then establish an inclusion theorem concerning the relation between these two beautiful and definitively useful concepts. Also, based upon our proposed ideas, we demonstrate new thoughtful approximation of Korovkin-type theorems for a martingale sequence over a Banach space. Moreover, we establish that our theorems effectively extend and improve most (if not all) of the previously existing outcomes (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.

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Section: Introductionmentioning

confidence: 99%

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

Jena

Paikray

2022

AMI

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“…Then, it has been addressed under various titles including Fourier analysis, Ergodic theory, Number theory, Turnpike theory, Measure theory, Trigonometric series, and Banach spaces. The concept was later applied to summability theory by various authors such as Çinar et al ( [5,6]), Çolak [7], Connor [8], Fridy [9], Altay et al [10], Garcia and Kama [11], ), Kucukaslan and Yılmazturk ([16,17]), Šalát [18], Ercan et al ( [19][20][21]), and Parida et al [22], and this concept has been extended to sequence spaces, accordingly, to the notions such as summability theory.…”

Section: Introductionmentioning

confidence: 99%

Examination of Generalized Statistical Convergence of Order $α$ on Time Scales

Çınar

Işık

2022

Journal of Function Spaces

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In this paper, we introduce the concepts of deferred statistical convergence of order α and strongly deferred Cesàro summable functions (real valued) of order α on time scales and give some relationships between deferred statistical convergence of order α and strongly deferred Cesàro summable functions (real valued) of order α on time scales.

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“…Moreover, the introduction of statistical probability convergence has enhanced the glory of this development. For some recent research works in this direction, see previous studies 3–15 …”

Section: Introductionmentioning

confidence: 99%

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 deferred Cesàro equi-statistical convergence and analogous approximation theorems

Cited by 13 publications

References 21 publications

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

Examination of Generalized Statistical Convergence of Order $α$ on Time Scales

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

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Generalized deferred Cesàro equi-statistical convergence and analogous approximation theorems

Cited by 13 publications

References 21 publications

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

A certain class of statistical convergence of martingale sequences and its applications to Korovkin-type approximation

Examination of Generalized Statistical Convergence of Order α on Time Scales

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

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Product

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Examination of Generalized Statistical Convergence of Order $α$ on Time Scales