Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

Srivástava, H. M.; Jena, Bidu Bhusan; Paikray, Susanta Kumar

doi:10.3390/axioms10030229

Cited by 8 publications

(6 citation statements)

References 20 publications

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“…In the year 2020, Jena and Paikray 8 established product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin‐type theorems. In the year 2021, Srivastava et al 9 introduced the concepts statistical convergence of Riemann and Lebesgue integrable sequence of functions to prove some Korovkin‐type approximation theorems. Recently, Srivastava et al 10 used deferred Cesàro statistical convergence and statistical product convergence of martingale sequences to establish some Korovkin‐type approximation theorems.…”

Section: Extension Towards Statistical Convergencementioning

confidence: 99%

Uniform convergence of Fourier series via deferred Cesàro mean and its applications

Jena

Paikray

Mursaleen

2022

Math Methods in App Sciences

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In this article, we found an undesirable feature in the theory of summability; that is, the Fourier series of 2π$$ 2\pi $$‐periodic functions is uniformly convergent to the functions via the Ceàsro mean. However, it does not preserve the uniform convergence for the arbitrary periodic functions. To overcome this limitation, the objective of the paper is to introduce and study the notion of deferred Ceàsro mean based on the q$$ q $$ integers for the Fourier series of arbitrary periodic functions. We further study the behavior of our proposed method under both ordinary and statistical versions of convergence and accordingly establish two Korovkin‐type approximation theorems for trigonometric test functions. Finally, we also provide some examples with geometrical illustrations in support of the effectiveness of our results established in this work.

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Section: Extension Towards Statistical Convergencementioning

confidence: 99%

Uniform convergence of Fourier series via deferred Cesàro mean and its applications

Jena

Paikray

Mursaleen

2022

Math Methods in App Sciences

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“…Remark 3. Motivated by some recently published results by Jena et al [32] and Srivastava et al [33], we choose to draw the attention of the interested readers toward the potential for further research associated with the analogous notion of statistical Lebesgue-measurable sequences of functions.…”

Section: Concluding Remarks and Directions For Further Researchmentioning

confidence: 99%

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

Srivástava

Jena

2022

Axioms

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Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.

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“…We may write it as g m ∈ L(Z, ρ). To know more about statistical Riemann integral and statistical Lebesgue integral, see [7,20].…”

Section: Introductionmentioning

confidence: 99%

Applications of statistical Riemann and Lebesgue integrability of sequence of functions

Raj¹,

Sharma²

2023

J. Nonlinear Sci. Appl.

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In the present work, we propose to investigate statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability by means of deferred N örlund and deferred Riesz mean. We discuss some fundamental theorems connecting these concepts with examples. As an application to our newly formed sequences, we introduce Korovkin-type approximation theorems with relevant example for positive linear operators by using Meyer-K önig and Zeller operators to exhibit the effectiveness of our findings.

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Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

Cited by 8 publications

References 20 publications

Uniform convergence of Fourier series via deferred Cesàro mean and its applications

Uniform convergence of Fourier series via deferred Cesàro mean and its applications

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

Applications of statistical Riemann and Lebesgue integrability of sequence of functions

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