Study of sensitivity of parameters of Bernstein-Stancu operators

TL;DR

This paper analyzes how the parameters  and  in Bernstein-Stancu operators influence their approximation behavior, providing convergence proofs, graphical comparisons, and insights into parameter tuning for improved pointwise approximation.

Contribution

It offers a detailed study of parameter sensitivity in Bernstein-Stancu operators, including convergence proofs and methods for optimizing approximation at specific points.

Findings

Parameters  and  affect approximation quality.

Graphical comparisons illustrate node behavior.

Optimal / ratio improves pointwise approximation.

Abstract

This paper is aimed at studying sensitivity of parameters \alpha and \beta appearing in the operators introduced by D.D. Stancu [11] in 1969. Results are established on the behavior the nodes used in Bernstein-Stancu polynomials and the nodes used in Bernstein polynomials and graphical presentations of them are generated. Alternate proof of uniform convergence of Bernstein-Stancu operators and an upper bound estimation are derived. It is also established that the parameters \alpha and \beta in Bernstein-Stancu polynomials can be used to get better approximation at a point x = (\alpha)/(\beta) in [0,1] to the Bernstein polynomials.