Comput. Methods Funct. Theory (2018) 18:361–388
Bernstein–Walsh Theory Associated to Convex Bodies
and Applications to Multivariate Approximation Theory
· N. Levenberg
Received: 2 March 2017 / Revised: 26 September 2017 / Accepted: 27 September 2017 /
Published online: 24 October 2017
© Springer-Verlag GmbH Germany 2017
Abstract We prove a version of the Bernstein–Walsh theorem on uniform polynomial
approximation of holomorphic functions on compact sets in several complex variables.
Here we consider subclasses of the full polynomial space associated to a convex
body P. As a consequence, we validate and clarify some observations of Trefethen in
multivariate approximation theory.
Keywords Convex body · Bernstein–Walsh · Multivariate approximation
Mathematics Subject Classiﬁcation 32U15 · 32U20 · 41A10
A standard theorem in several complex variables, quantifying the classical Oka–Weil
theorem on polynomial approximation—which itself is the multivariate version of the
classical Runge theorem for polynomial approximation in the complex plane—is the
Communicated by Doron Lubinsky.
N. Levenberg has been supported by Simons Foundation Grant no. 354549.
University of Verona, Verona, Italy
Indiana University, Bloomington, IN 47405, USA