By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set can be approximated uniformly on K by complex polynomials. The famous Bernstein-Walsh-Siciak theorem specifies the Oka-Weil result: it states that the distance (in the supremum norm on K) of f to the space of complex polynomials of degree at most n tends to zero not slower than the sequence M(f)ρ(f)ⁿ for some M(f) > 0 and ρ(f) ∈ (0,1). The aim of this note is to deduce the uniform version, sometimes called family version, of the Bernstein-Walsh-Siciak theorem, which is due to Pleśniak, directly from its classical (weak) form. Our method, involving the Baire category theorem in Banach spaces, appears to be useful also in a completely different context, concerning Łojasiewicz’s inequality.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-1-4,
author = {Rafa\l\ Pierzcha\l a},
title = {On the Bernstein-Walsh-Siciak theorem},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {55-63},
zbl = {1269.32007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-1-4}
}
Rafał Pierzchała. On the Bernstein-Walsh-Siciak theorem. Studia Mathematica, Tome 209 (2012) pp. 55-63. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-1-4/