On the Bernstein–Walsh–Siciak theorem
Volume 212 / 2012
Abstract
By the Oka–Weil theorem, each holomorphic function $f$ in a neighbourhood of a compact polynomially convex set $K\subset\mathbb{C}^N$ 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 $M(f)\rho (f)^n$ for some $M(f)>0$ and $\rho (f) \in (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.
