A generalisation of de la Vallée-Poussin procedure to multivariate approximations
Nadezda Sukhorukova, Julien Ugon
TL;DR
The paper addresses extending minimax Chebyshev approximation to multivariate settings by generalizing the de la Vallée-Poussin procedure to arbitrary basis functions beyond monomials. It shows the objective Psi(A) is convex because L(A,x) is affine in A and uses subgradient analysis; optimality is linked to the intersection of convex hulls of positive and negative deviation directions, with Radon-type partitions ensuring existence when the dimension satisfies n >= d. The authors propose a three-step extension of the classical univariate procedure, providing a constructive framework for multidimensional Chebyshev interpolation modelling functions and proving that appropriate exchanges strictly increase the deviation. They discuss maintaining nonsingular bases and outline future work on singular bases and weakening geometric assumptions. Overall, the work broadens the applicability of extremal Chebyshev approximation to higher dimensions and general basis families.
Abstract
The theory of Chebyshev approximation has been extensively studied. In most cases, the optimality conditions are based on the notion of alternance or alternating sequence (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical de la Vallée-Poussin procedure. In this paper we demonstrate that under certain assumptions the classical de la Vallée-Poussin procedure, developed for univariate polynomial approximation, can be extended to the case of multivariate approximation. The corresponding basis functions are not restricted to be monomials.