Approximation polynomiale et extension holomorphe avec croissance sur une variété algébrique
Tom 63 / 1996
Annales Polonici Mathematici 63 (1996), 35-50 DOI: 10.4064/ap-63-1-35-50
We first give a general growth version of the theorem of Bernstein-Walsh-Siciak concerning the rate of convergence of the best polynomial approximation of holomorphic functions on a polynomially convex compact subset of an affine algebraic manifold. This can be considered as a quantitative version of the well known approximation theorem of Oka-Weil. Then we give two applications of this theorem. The first one is a generalization to several variables of Winiarski's theorem relating the growth of an entire function to the rate of convergence of its best polynomial approximation; the second application concerns the extension with growth of an entire function from an algebraic submanifold to the whole space.