A characterization of holomorphic germs on compact perfect sets

Graciela Carboni; Angel Rafael Larotonda

Displaying similar documents to “A characterization of holomorphic germs on compact perfect sets”

Holomorphic series expansion of functions of Carleman type

Taib Belghiti (2004)

Annales Polonici Mathematici

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Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying $s u p_{z \in D ₙ} | f ₙ (z) | \leq C ϱ ⁿ$ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f. ...

Pervasive algebras on planar compacts

Jan Čerych (1999)

Commentationes Mathematicae Universitatis Carolinae

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We characterize compact sets $X$ in the Riemann sphere $𝕊$ not separating $𝕊$ for which the algebra $A (X)$ of all functions continuous on $𝕊$ and holomorphic on $𝕊 ∖ X$ , restricted to the set $X$ , is pervasive on $X$ .

Converging semigroups of holomorphic maps

Marco Abate (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we study the semigroups $\Phi:\mathbb{R}^{+}\rightarrow Hol(D,D)$ of holomorphic maps of a strictly convex domain $D\subset\mathbf{C}^{n}$ into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map $h:D\rightarrow\mathbf{C}^{n}$ .

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg, Mark Elin, David Shoikhet (2005)

Studia Mathematica

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The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: $| \sum_{n = 0}^{\infty} a ₙ z ⁿ | < 1$ , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $| \sum_{n = 0}^{k} a ₙ z ⁿ | < 1$ , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are...

Variations of complex structures on an open Riemann surface

M. S. Narasimhan (1961)

Annales de l'institut Fourier

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Soit $U_{1}$ un ouvert dans $C^{m}$ . Soit $π_{1} : S \to U_{1}$ une famille holomorphe de structures complexes sur une surface de Riemann non-compacte $M$ , avec $S_{t_{0}} = π_{1}^{- 1} (t_{0}) = M$ . ( $S = S (M \times U_{1})$ est une structure complexe sur le produit différentiable $M \times U_{1}$ ). Soit $M_{1}$ un domaine relativement compact dans $M$ . On démontre : pour tout voisinage de Stein $U$ de $t_{0}$ , assez petit, la famille $π_{1} : S (M_{1} \times U) \to U$ est isomorphe à la famille $π : Ω \to π (Ω)$ , où $Ω$ est un de la variété produit $M \times C^{m}$ , $π$ étant la projection $M \times C^{m} \to C^{m}$ . On donne aussi un résultat analogue pour le cas des variations différentiables. ...

A set on which the local Łojasiewicz exponent is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F = (f ₁, . . ., f ₘ) : U \to ℂ^{m}$ , F(0) = 0, the Łojasiewicz exponent ₀(F) is attained on the set z ∈ U: f₁(z)·...·fₘ(z) = 0.

Contracting rigid germs in higher dimensions

Matteo Ruggiero (2013)

Annales de l’institut Fourier

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Following Favre, we define a holomorphic germ $f : (ℂ^{d}, 0) \to (ℂ^{d}, 0)$ to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of $f$ and its linear action on the fundamental group of the complement of the critical set.

Converging semigroups of holomorphic maps

Marco Abate (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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On a class of inner maps

Edoardo Vesentini (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let $f$ be a continuous map of the closure $\bar{Δ}$ of the open unit disc $Δ$ of $C$ into a unital associative Banach algebra $A$ , whose restriction to $Δ$ is holomorphic, and which satisfies the condition whereby $0 \notin σ (f (z)) \subset \bar{Δ}$ for all $z \in Δ$ and $σ (f (z)) \subset \partial Δ$ whenever $z \in \partial Δ$ (where $σ (x)$ is the spectrum of any $x \in A$ ). One of the basic results of the present paper is that $f$ is , that is to say, $σ (f (z))$ is then a compact subset of $\partial Δ$ that does not depend on $z$ for all $z \in \bar{Δ}$ . This fact will be applied to holomorphic self-maps of the open unit ball of some...

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let $D_{j} \subset ℂ^{k_{j}}$ be a pseudoconvex domain and let $A_{j} \subset D_{j}$ be a locally pluriregular set, j = 1,...,N. Put $X : = ⋃_{j = 1}^{N} A ₁ \times . . . \times A_{j - 1} \times D_{j} \times A_{j + 1} \times . . . \times A_{N} \subset ℂ^{k ₁ + . . . + k_{N}}$ . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f ̂ |}_{X ∖ M} = f$ . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...