Enhanced Approximation Techniques: Stancu-type (λ,μ)-Bernstein-Kantorovich Operators

Abstract

The principal aim of this study is to investigate a range of approximation characteristics exhibited by Stancu-type (λ, μ)-Bernstein-Kantorovich operators denoted as Fλ,μ r,α,β(ϕ; z). In the initial stage, we conduct a comprehensive examination of diverse moment estimates pertaining to operators Fλ,μ r,α,β(ϕ; z). Subsequently, we delve into the exploration of several facets of direct results, including the order of convergence with respect to the usual modulus of continuity, Lipschitz-type continuous functions, and the Peetre’s K-functional. Furthermore, in order to gain insights into the asymptotic characteristics of the operators Fλ,μ r,α,β(ϕ; z), we derive a Voronovskaya-type asymptotic theorem. Additionally, we provide an analysis of the A-statistical convergence behavior and pointwise estimates associated with the operators Fλ,μ r,α,β(ϕ; z). Finally, we have included graphical representations and numerical error value tables to demonstrate the efficiency and accuracy of our proposed operator. Our analysis reveals that our operator yields significantly superior approximation results compared to certain other linear positive operators documented in the existing literature. Keywords and