Approximation Properties of $(\lambda,\mu)$-Bernstein Operators

Abstract

This paper introduces a novel $(\lambda, \mu)$-Bernstein operator $B_{n}^{(\lambda, \mu)}(f; x)$, derived from a newly developed Bézier basis with the shape parameters $\lambda$ and $\mu$. This operator generalizes the classical Bernstein operator while preserving its fundamental approximation properties and providing enhanced flexibility in function approximation. Fundamental theoretical results are rigorously established, including explicit formulas for moments, a Korovkin-type approximation theorem, and convergence properties for Lipschitz continuous functions. Specifically, a Voronovskaja-type asymptotic expansion is derived to rigorously describe the operator's asymptotic behavior.

Numerical experiments validate the theoretical findings, demonstrating that the proposed operators achieve lower approximation errors than both the classical Bernstein and $\lambda$-Bernstein operators. These results highlight the adaptability and improved approximation capabilities of the $(\lambda, \mu)$-Bernstein operators, making them a promising tool in approximation theory and computational mathematics.