On the Bernstein-Walsh-Siciak theorem

• Autorzy: Rafał Pierzchała
• Języki publikacji: EN

Abstrakty

EN

By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set $K ⊂ ℂ^{N}$ 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)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.

Kategorie tematyczne

• 32B20: Semi-analytic sets and subanalytic sets
• 41A10: Approximation by polynomials
• 41A25: Rate of convergence, degree of approximation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2012
• Tom: 212
• Numer: 1
• Strony: 55-63

Daty

wydano

2012

Twórcy

autor

Rafał Pierzchała

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

Identyfikatory

DOI

10.4064/sm212-1-4