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## Studia Mathematica

2012 | 212 | 1 | 55-63
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### On the Bernstein-Walsh-Siciak theorem

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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.
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Tom
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55-63
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wydano
2012
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• Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
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