Optimization-aided construction of multivariate Chebyshev polynomials

IF 0.9 3区 数学 Q2 MATHEMATICS
M. Dressler , S. Foucart , M. Joldes , E. de Klerk , J.B. Lasserre , Y. Xu
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引用次数: 0

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 x12x22x3 on the euclidean ball and for the monomial x12x2x3 on the simplex.
优化辅助构建多元切比雪夫多项式
本文关注的是将第一类单变量切比雪夫多项式扩展到多变量环境,即通过相对于统一规范的低度多项式来追寻特定单项式的最佳近似值。利用 Moment-SOS 层次结构,我们设计了一种基于半定量编程的通用程序,用于计算此类最佳近似值以及相关签名。在三个变量中应用这一程序,就能得出欧几里得球、简单面和交叉多面体上六度以内所有单项式的最佳近似误差值。此外,在数值实验的启发下,我们还得到了切比雪夫多项式在两种情况下的明确表达式,即欧几里得球上的单项式 x12x22x3 和单纯形上的单项式 x12x2x3。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.90
自引率
11.10%
发文量
55
审稿时长
6-12 weeks
期刊介绍: The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.
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