On the Bernstein–Walsh–Siciak theorem

Rafał Pierzchała¹

¹ Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Studia Mathematica, Tome 212 (2012) no. 1, pp. 55-63

Résumé

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.

Détail

DOI : 10.4064/sm212-1-4

Mots-clés : oka weil theorem each holomorphic function neighbourhood compact polynomially convex set subset mathbb approximated uniformly complex polynomials famous bernstein walsh siciak theorem specifies oka weil result states distance supremum norm space complex polynomials degree tends zero slower sequence rho rho note deduce uniform version sometimes called family version bernstein walsh siciak theorem which due ple niak directly its classical weak form method involving baire category theorem banach spaces appears useful completely different context concerning ojasiewiczs inequality