Optimization-Aided Construction of Multivariate Chebyshev Polynomials
Résumé
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 mononials 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.
Mots clés
Optimization and Control (math.OC)
Numerical Analysis (math.NA)
FOS: Mathematics
41A10
65D15
90C22
best approximation Chebyshev polynomials sum of squares method of moments semidefinite programming AMS classification: 41A10 65D15 90C22 Optimization-Aided Construction of Multivariate Chebyshev
best approximation
Chebyshev polynomials
sum of squares
method of moments
semidefinite programming AMS classification: 41A10
90C22 Optimization-Aided Construction of Multivariate Chebyshev
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