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Optimization-Aided Construction of Multivariate Chebyshev Polynomials

Mareike Dressler, Simon Foucart, Mioara Joldes, Etienne de Klerk, Jean Bernard Lasserre, Yuan Xu

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.

Optimization-Aided Construction of Multivariate Chebyshev Polynomials

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 on the euclidean ball and for the monomial on the simplex.
Paper Structure (15 sections, 10 theorems, 51 equations, 3 tables)

This paper contains 15 sections, 10 theorems, 51 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Prior Knowledge
  3. Characterization via signatures
  4. Simple reductions
  5. Known multivariate Chebyshev polynomials
  6. The hypercube.
  7. The euclidean ball.
  8. The simplex.
  9. Description of the Computational Procedure
  10. New Explicit Results
  11. The euclidean ball
  12. The cross-polytope
  13. The simplex
  14. Conclusion
  15. Acknowledgment.

Key Result

Theorem 2

An element $v^* \in \mathcal{V}$ is a best approximant to $f \in C(\Omega)$ from $\mathcal{V}$ if and only if there exists an extremal signature $\sigma$ for $\mathcal{V}$ associated with $f-v^*$. Moreover, the support of such a signature can be chosen to have size $\le \dim(\mathcal{V})+1$.

Theorems & Definitions (16)

  • Definition 1
  • Theorem 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • ...and 6 more