On the Bernstein-Walsh-Siciak theorem

Rafał Pierzchała. "On the Bernstein-Walsh-Siciak theorem." Studia Mathematica 212.1 (2012): 55-63. <http://eudml.org/doc/285387>.

@article{RafałPierzchała2012,
abstract = {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.},
author = {Rafał Pierzchała},
journal = {Studia Mathematica},
keywords = {polynomial approximation; Oka-Weil theorem; Bernstein-Walsh-Siciak theorem; Łojasiewicz inequality; Baire category theorem},
language = {eng},
number = {1},
pages = {55-63},
title = {On the Bernstein-Walsh-Siciak theorem},
url = {http://eudml.org/doc/285387},
volume = {212},
year = {2012},
}

TY - JOUR
AU - Rafał Pierzchała
TI - On the Bernstein-Walsh-Siciak theorem
JO - Studia Mathematica
PY - 2012
VL - 212
IS - 1
SP - 55
EP - 63
AB - 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.
LA - eng
KW - polynomial approximation; Oka-Weil theorem; Bernstein-Walsh-Siciak theorem; Łojasiewicz inequality; Baire category theorem
UR - http://eudml.org/doc/285387
ER -