Chebyshev multivariate polynomial approximation and point reduction procedure

TL;DR

This paper extends Chebyshev approximation theory to multivariate functions, providing geometric interpretations, a fast necessary condition verification algorithm, and a generalized alternating sequence for optimality conditions.

Contribution

It introduces a geometric interpretation of multivariate optimality conditions and develops a fast algorithm for verifying necessary conditions, along with a generalized alternating sequence.

Findings

The algorithm significantly speeds up necessary condition verification.

The geometric interpretation applies to various basis functions.

A generalized alternating sequence for multivariate polynomials is proposed.

Abstract

The theory of Chebyshev (uniform) approximation for univariate polynomial and piecewise polynomial functions has been studied for decades. The optimality conditions are based on the notion of alternating sequence. However, the extension the notion of alternating sequence to the case of multivariate functions is not trivial. The contribution of this paper is two-fold. First of all, we give a geometrical interpretation of the necessary and sufficient optimality condition for multivariate approximation. These optimality conditions are not limited to the case polynomial approximation, where the basis functions are monomials. Second, we develop an algorithm for fast necessary optimality conditions verifications (polynomial case only). Although, this procedure only verifies the necessity, it is much faster than the necessary and sufficient conditions verification. This procedure is based on a point reduction procedure and resembles the univariate alternating sequence based optimality conditions. In the case of univariate approximation, however, these conditions are both necessary and sufficient. Third, we propose a procedure for necessary and sufficient optimality conditions verification that is based on a generalisation of the notion of alternating sequence to the case of multivariate polynomials.