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

Hari Mohan Srivastava,Susanta Kumar Paikray,Bidu Bhusan Jena

doi:10.3390/axioms11030128

Hari Mohan Srivastava, Susanta Kumar Paikray + Show 1 more

Open Access

https://doi.org/10.3390/axioms11030128

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Journal: Axioms	Publication Date: Mar 12, 2022
Citations: 1	License type: CC BY 4.0

Affiliation: Veer Surendra Sai University of Technology

Abstract

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.

Highlights

Motivated mainly by the above-mentioned investigations and developments, we introduce and study the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability of sequences of real-valued functions
Based upon the core concept of statistical Lebesgue-measurable sequences of functions, we suggest some possible directions for future research on this topic in the concluding section of our study
Let Gj : C[0, 1] → C[0, 1] be a sequence of positive linear operators

Summary

The resulting partition P is then given by

A sequence (hk)k∈N of functions is Riemann-integrable to h on [a, b] if, for each > 0, there exists σ > 0 such that, for any tagged partition P of [a, b] with P < σ , we have. A sequence (hk)k∈N of functions is statistically Riemann-integrable to h on [a, b] if, for every > 0 and for each x ∈ [a, b], there exists σ > 0, and for any tagged partition P of [a, b]. A sequence (hk)k∈N of functions is said to be deferred weighted statistically Riemannintegrable to h on [a, b] if, for all > 0, there exists σ > 0, and for any tagged partition P of [a, b]. A sequence (hk)k∈N of functions is said to statistically deferred weighted Riemann summable to h on [a, b] if, for all > 0 ∃ σ > 0 and for any tagged partition P of [a, b] with.

We thus obtain

Riemann sum

We thus find that

We also know that