Convergence of the summation-integral type operators via statistically

TL;DR

This paper studies the statistical convergence properties of summation-integral operators, establishing convergence theorems, rates, and extensions to bivariate functions, supported by graphical illustrations and comparisons.

Contribution

It introduces new statistical convergence results for summation-integral operators, including weighted and bivariate cases, with graphical and comparative analyses.

Findings

Proved statistical convergence theorems using Korovkin's theorem.

Established rates of convergence via modulus of continuity and Lipschitz class.

Demonstrated convergence through graphical representations and comparisons.

Abstract

Our main aim is to investigate the approximation properties for the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of approximation is determined. Also using weight function, the weighted statistical convergence theorem with the help of Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class is obtained. To support the convergence results of the proposed operators to the function, graphical representations take place and a comparison is shown with Sz\'asz-Mirakjan-Kantorovich operators through examples. The last section deals with, a bivariate extension of the proposed operators to study the rate of convergence for the function of two variables, additionally, the convergence of the bivariate operators is shown graphically.