Optimization-aided construction of multivariate Chebyshev polynomials

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 versat...

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Bibliographic Details
Published in:Journal of approximation theory 2025-01, Vol.305, p.106116, Article 106116
Main Authors: Dressler, M., Foucart, S., Joldes, M., de Klerk, E., Lasserre, J.B., Xu, Y.
Format: Article
Language:English
Subjects:
Best approximation
Chebyshev polynomials
Method of moments
Semidefinite programming
Sum of squares
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Summary: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 x12x22x3 on the euclidean ball and for the monomial x12x2x3 on the simplex.
ISSN:0021-9045
DOI:10.1016/j.jat.2024.106116