Multivariate polynomial approximation in the hypercube

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 $x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.