Approximation properties by some modified Szasz-Mirakjan-Kantorovich operators

TL;DR

This paper investigates the approximation capabilities of modified Szasz-Mirakjan-Kantorovich operators using various smoothness measures, establishing convergence rates and quantitative theorems with illustrative examples.

Contribution

It introduces new local approximation results for these operators using Lipschitz and Ditzian-Totik moduli, along with convergence rates and quantitative theorems.

Findings

Established convergence rates in terms of Lipschitz and Ditzian-Totik moduli.

Proved quantitative Voronovskaya and Grüss type theorems for the operators.

Provided graphical examples supporting the theoretical results.

Abstract

The present article deals with the local approximation results by means of Lipschitz maximal function, Ditzian-Totik modulus of smoothness and Lipschitz type space having two parameters for the summation-integral type operators defined by Mishra and Yadav (Tbilisi Mathematical Journal. 11(3), (2018), 175-91). Further, we determine the rate of convergence in the term of the with derivative of bounded variation and for the quantitative means of the defined operators, we establish the quantitative Voronovskaya type and Gr$\ddot{\text{u}}$ss type theorems. Moreover, the examples are given with graphical representation to support the main results.