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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.

Bernstein–Walsh Theory Associated to Convex Bodies and Applications to Multivariate Approximation Theory