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Wiener's type regularity criteria on the complex plane

Józef Siciak — 1997

Annales Polonici Mathematici

We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that d ( t - 1 E ∩ | z - a | ≤ 1 ) ≥ c o n s t > 0 for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is Hölder continuous....

Sets in ℂ N with vanishing global extremal function and polynomial approximation

Józef Siciak — 2011

Annales de la faculté des sciences de Toulouse Mathématiques

Let Γ be a non-pluripolar set in ℂ N . Let f be a function holomorphic in a connected open neighborhood G of Γ . Let { P n } be a sequence of polynomials with deg P n ≤ d n ( d n &lt; d n + 1 ) such that lim sup n → ∞ | f ( z ) - P n ( z ) | 1 / d n &lt; 1 , z ∈ Γ . We show that if lim sup n → ∞ | P n ( z ) | 1 / d n ≤ 1 , z ∈ E , where E is a set in ℂ N such that the global extremal function V E ≡ 0 in ℂ N , then the maximal domain of existence G f of f is one-sheeted, and lim sup n → ∞ ∥ f - P n ∥ K 1 d n &lt; 1 for every compact set K ⊂ G f . If, moreover, the sequence { d n + 1 / d n } is bounded then G f = ℂ N . If E is a closed set in ℂ N then...

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