Modified Bernstein Polynomials and Jacobi Polynomials in q-Calculus

TL;DR

This paper generalizes modified Bernstein polynomials for Jacobi weights using q-calculus, extending their properties and introducing q-analogues of classical approximation results and eigenvectors.

Contribution

It introduces a new q-analogue of modified Bernstein polynomials for Jacobi weights, expanding their theoretical properties and applications.

Findings

Extended approximation properties to q-analogues

Proved convergence and shape-preserving properties in q-calculus

Analyzed eigenvectors as q-extensions of Jacobi polynomials

Abstract

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the $q$-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in $]0,1[$. Numerous properties of the modified Bernstein Polynomials are extended to their $q$-analogues: simultaneous approximation, pointwise convergence even for unbounded functions, shape-preserving property, Voronovskaya theorem, self-adjointness. Some properties of the eigenvectors, which are $q$-extensions of Jacobi polynomials, are given.