Multivariate polynomial approximation in the hypercube

Trefethen, Lloyd

doi:10.1090/proc/13623

Multivariate polynomial approximation in the hypercube
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by Lloyd N. Trefethen PDF

Proc. Amer. Math. Soc. 145 (2017), 4837-4844 Request permission

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 Euclidean degree, defined in terms of the 2-norm rather than the 1-norm of the exponent vector $\mathbf {k}$ of a monomial $x_1^{k_1}\cdots \kern .8pt x_s^{k_s}$.

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