Approximations on Stancu variant of Sz\'asz-Mirakjan-Kantorovich type operators

TL;DR

This paper analyzes the approximation properties and convergence behavior of a modified Stancu variant of Szász-Mirakjan-Kantorovich operators, providing theoretical results and numerical illustrations.

Contribution

It introduces and studies a new variant of these operators, establishing their approximation order, convergence rates, and asymptotic behavior in various function spaces.

Findings

Determined the order of approximation using modulus of continuity

Established convergence rates in Lipschitz and weighted spaces

Proved Voronovskaya-type theorems for the operators

Abstract

This paper discusses the properties of a modified version of the Stancu variant Sz\'asz-Mirakjan Kantorovich type operators. We determine the order of approximation in terms of the modulus of continuity and second-order of smoothness, and we obtain the rate of convergence using the Lipschitz space. We also establish a relation using the Peetre-$K$ functional for the proposed operators. An asymptotic formula for these operators is also established to understand the asymptotic behavior. Along with these discussions, we give some convergence properties in weighted spaces using the weight function. Moving ahead in weighted spaces, the Quantitative Voronovskaya-type and Gr\"{u}ss Voronovskaya-type theorems are proved using the weighted modulus of continuity. Furthermore, we determine the rate of convergence of the proposed operators in terms of functions with derivatives of bounded variations. Finally, we provide graphical representations and numerical analysis for our theoretical findings.