EUDML

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 $d (t^{- 1} E \cap | z - a | \leq 1) \geq c o n s t > 0$ 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....

Singular sets of separately analytic functions

Józef Siciak — 1990

Colloquium Mathematicae

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 $deg P_{n} \leq d_{n} (d_{n} < d_{n + 1})$ such that $\underset{n \to \infty}{lim sup} {| f (z) - P_{n} (z) |}^{1 / d_{n}} < 1, z \in Γ .$ We show that if $\underset{n \to \infty}{lim sup} {| P_{n} (z) |}^{1 / d_{n}} \leq 1, z \in E,$ where $E$ is a set in $ℂ^{N}$ such that the global extremal function $V_{E} \equiv 0$ in $ℂ^{N}$ , then the maximal domain of existence $G_{f}$ of $f$ is one-sheeted, and $\underset{n \to \infty}{lim sup} {∥ f - P_{n} ∥}_{K}^{\frac{1}{d_{n}}} < 1$ for every compact set $K \subset G_{f}$ . 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...

Families of polynomials and determining measures

Jozef Siciak — 1988

Annales de la Faculté des sciences de Toulouse : Mathématiques

Some gap power series in multidimensional setting

Józef Siciak — 2011

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

We study extensions of classical theorems on gap power series of a complex variable to the multidimensional case.

Polynomials and multilinear mappings in topological vector-spaces

Jacek Bochnak; Józef Siciak — 1971

Studia Mathematica

Analytic functions in topological vector-spaces

Jacek Bochnak; Józef Siciak — 1971

Studia Mathematica

Bounded holomorphic functions with multiple sheeted pluripolar hulls

Armen Edigarian; Józef Siciak; Włodzimierz Zwonek — 2006

Studia Mathematica

We describe compact subsets K of ∂𝔻 and ℝ admitting holomorphic functions f with the domains of existence equal to ℂ∖K and such that the pluripolar hulls of their graphs are infinitely sheeted. The paper is motivated by a recent paper of Poletsky and Wiegerinck.

On the product property for the transfinite diameter

Zbigniew Błocki; Armen Edigarian; Józef Siciak — 2011

Annales Polonici Mathematici

We give a pluripotential-theoretic proof of the product property for the transfinite diameter originally shown by Bloom and Calvi. The main tool is the Rumely formula expressing the transfinite diameter in terms of the global extremal function.

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Approximation by transcendental polynomials

Extremal plurisubharmonic functions in $C^{N}$

On series of homogeneous polynomials and their partial sums

Some gap power series in multidimensional setting

Wiener's type regularity criteria on the complex plane

Singular sets of separately analytic functions

Sets in $ℂ^{N}$ with vanishing global extremal function and polynomial approximation

Families of polynomials and determining measures

Some gap power series in multidimensional setting

Polynomials and multilinear mappings in topological vector-spaces

Analytic functions in topological vector-spaces

Bounded holomorphic functions with multiple sheeted pluripolar hulls

On the product property for the transfinite diameter

A note on pluripolar extensions of univalent functions.

Holomorphic functions with singularities on algebraic sets.

A note on Alexander's theorem.