# Computation of Chebyshev Polynomials for Union of Intervals

## TL;DR

This paper introduces numerical methods based on semidefinite programming to compute Chebyshev polynomials of the first and second kind for sets that are finite unions of intervals, with applications in approximation theory.

## Contribution

It develops novel semidefinite programming procedures for calculating Chebyshev polynomials on unions of intervals, incorporating root constraints and leveraging polynomial nonnegativity and moments.

## Key findings

- Effective numerical procedures for Chebyshev polynomial computation.
- Ability to incorporate root location constraints.
- Application of semidefinite programming to polynomial approximation.

## Abstract

Chebyshev polynomials of the first and second kind for a set K are monic polynomials with minimal L $\infty$-and L 1-norm on K, respectively. This articles presents numerical procedures based on semidefinite programming to compute these polynomials in case K is a finite union of compact intervals. For Chebyshev polynomials of the first kind, the procedure makes use of a characterization of polynomial nonnegativity. It can incorporate additional constraints, e.g. that all the roots of the polynomial lie in K. For Chebyshev polynomials of the second kind, the procedure exploits the method of moments. Key words and phrases: Chebyshev polynomials of the first kind, Chebyshev polynomials of the second kind, nonnegative polynomials, method of moments, semidefinite programming.