Characterization of the generalized Chebyshev-type polynomials of first kind

TL;DR

This paper characterizes generalized Chebyshev-type polynomials of the first kind, providing a closed form in terms of Bernstein polynomials and exploring their integration properties, advancing approximation theory techniques.

Contribution

It introduces a new characterization and closed-form expression of generalized Chebyshev-type polynomials of the first kind in terms of Bernstein polynomials.

Findings

Closed-form expression of generalized Chebyshev-type polynomials in Bernstein form

New insights into the integration of these polynomials with Bernstein polynomials

Enhanced tools for approximation theory using orthogonal polynomials

Abstract

Orthogonal polynomials have very useful properties in the solution of mathematical problems, so recent years have seen a great deal in the field of approximation theory using orthogonal polynomials. In this paper, we characterize the generalized Chebyshev-type polynomials of the first kind $\mathscr{T}_{n}^{(M,N)}(x),$ then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials $B_{k}^{n}(x).$ We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.