Multivariate polynomial approximation in the hypercube

TL;DR

This paper establishes a theorem on approximating analytic functions in a hypercube using multivariate polynomials, revealing that geometric convergence depends on Euclidean degree rather than traditional polynomial degree.

Contribution

It introduces the concept of Euclidean degree for multivariate polynomial approximation, providing a new perspective on convergence rates in high-dimensional spaces.

Findings

Convergence rate is governed by Euclidean degree.

Euclidean degree differs from traditional degree in polynomial approximation.

Theorem applies to analytic functions in hypercubes.

Abstract

A theorem is proved concerning approximation of analytic functions by multivariate polynomials in the $s$-dimensional hypercube. The geometric convergence rate is determined not by the usual notion of degree of a multivariate polynomial, but by the {\it Euclidean degree,} defined in terms of the 2-norm rather than the 1-norm of the exponent vector $\bf k$ of a monomial $x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.