  In this paper we generalize to a certain class of Stein manifolds the
Bernstein-Walsh-Siciak theorem which describes the equivalence between possible
holomorphic continuation of a function $f$ defined on a compact set $K$ in
$\mathbb{C}^N$ to the rapidity of the best uniform approximation of $f$ on $K$
by polynomials. We also generalize Winiarski's theorem which relates the growth
rate of an entire function $f$ on $\mathbb{C}^N$ to its best uniform
approximation by polynomials on a compact set.
