Optimization-Aided Construction of Multivariate Chebyshev Polynomials

TL;DR

This paper extends univariate Chebyshev polynomials to multivariate cases using semidefinite programming, providing new approximation techniques and explicit polynomial expressions for complex monomials.

Contribution

It introduces a semidefinite-programming-based method for constructing multivariate Chebyshev polynomials and computes best approximation errors for various domains and monomials.

Findings

Computed best approximation errors for monomials up to degree six.

Derived explicit Chebyshev polynomial expressions for previously unresolved cases.

Validated the method on the Euclidean ball, simplex, and cross-polytope.

Abstract

This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a versatile semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean ball, the simplex, and the cross-polytope. Furthermore, inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before, namely for the monomial $x_1^2 x_2^2 x_3$ on the euclidean ball and for the monomial $x_1^2 x_2 x_3$ on the simplex.