Generalized Tschebyscheff of the second kind and Bernstein polynomials change of bases

TL;DR

This paper develops multiple basis representations for generalized Tschebyscheff polynomials of the second kind, including explicit change-of-basis matrices and formulas relating these polynomials to Bernstein basis polynomials.

Contribution

It introduces explicit change-of-basis matrices and formulas connecting generalized Tschebyscheff polynomials to Bernstein polynomials, expanding the tools for polynomial basis transformations.

Findings

Derived explicit change-of-basis matrices

Provided closed-form expressions for polynomial conversions

Enhanced understanding of polynomial basis relationships

Abstract

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. We build the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n.