Fractional $\alpha$-Bernstein-Kantorovich operators of order $\beta$: A new construction and approximation results

TL;DR

This paper introduces a new version of fractional $eta$-Bernstein-Kantorovich operators that improve function approximation, preserves linear functions, and extends to bivariate blending operators, with theoretical analysis and MATLAB visualizations.

Contribution

It presents a novel construction of fractional $eta$-Bernstein-Kantorovich operators, including their properties and bivariate extensions, advancing approximation theory methods.

Findings

Operators effectively approximate Lebesgue integrable functions.

Preservation of linear functions demonstrated.

Convergence and error bounds established with visual support.

Abstract

In the current article, we establish a distinct version of the operators defined by Berwal \emph{et al.}, which is the Kantorovich type modification of $\alpha$-Bernstein operators to approximate Lebesgue's integrable functions. We define its modification that can preserve the linear function and analyze its characteristics. Additionally, we construct the bivariate of blending type operators by Berwal \emph{et al.}. We analyze both its the convergence and error of approximation properties by using the conventional tools of approximation theory. Finally, we demonstrate our results by presenting examples that highlight graphical visuals using MATLAB.