Multivariate polynomial approximation in the hypercube
Provides a new theoretical result for approximation theory, relevant to researchers in multivariate polynomial approximation.
The paper proves a theorem on the approximation of analytic functions by multivariate polynomials in the hypercube, showing that the geometric convergence rate is determined by the Euclidean degree (2-norm of exponent vector) rather than the usual degree (1-norm).
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 $x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.