On the Bernstein–Walsh–Siciak theorem

Volume 212 / 2012

Rafał Pierzchała Studia Mathematica 212 (2012), 55-63 MSC: 32B20, 41A10, 41A25. DOI: 10.4064/sm212-1-4


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.


  • Rafał PierzchałaFaculty of Mathematics and Computer Science
    Jagiellonian University
    Łojasiewicza 6
    30-348 Kraków, Poland

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