On holomorphic continuation of functions along boundary sections

S. A. Imomkulov; J. U. Khujamov

Displaying similar documents to “On holomorphic continuation of functions along boundary sections”

Interpolation of bounded sequences

Francesc Tugores (2010)

Czechoslovak Mathematical Journal

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This paper deals with an interpolation problem in the open unit disc $𝔻$ of the complex plane. We characterize the sequences in a Stolz angle of $𝔻$ , verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on $𝔻$ , but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.

Norm and Taylor coefficients estimates of holomorphic functions in balls

Jacob Burbeam, Do Young Kwak (1991)

Annales Polonici Mathematici

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A classical result of Hardy and Littlewood states that if $f (z) = \sum_{m = 0}^{\infty} a_{m} z^{m}$ is in $H^{p}$ , 0 < p ≤ 2, of the unit disk of ℂ, then $\sum_{m = 0}^{\infty} {(m + 1)}^{p - 2} {| a_{m} |}^{p} \leq c_{p} ∥ f ∥_{p}^{p}$ where $c_{p}$ is a positive constant depending only on p. In this paper, we provide an extension of this result to Hardy and weighted Bergman spaces in the unit ball of $ℂ^{n}$ , and use this extension to study some related multiplier problems in $ℂ^{n}$ .

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. ...

A boundary cross theorem for separately holomorphic functions

Peter Pflug, Viêt-Anh Nguyên (2004)

Annales Polonici Mathematici

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Let D ⊂ ℂⁿ and $G \subset ℂ^{m}$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally ² smooth on A (resp. B). We shall determine the “envelope of holomorphy” X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this...

The exceptional sets for functions of the Bergman space in the unit ball

Piotr Jakóbczak (1993)

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

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Let $D$ be a domain in $C^{2}$ . Given $w \in C$ , set $D_{w} = \{z \in C ∣ (z, w) \in D\}$ . If $f$ is a holomorphic and square-integrable function in $D$ , then the set $E (D, f)$ of all $w$ such that $f (\cdot, w)$ is not square-integrable in $D_{w}$ has measure zero. We call this set the exceptional set for $f$ . In this Note we prove that whenever $0 < r < 1$ there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in $C^{2}$ such that $E (B, f)$ is the circle $C (0, r) = \{z \in C ∣ |z| = r\}$ .

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]. ...

Zeros of bounded holomorphic functions in strictly pseudoconvex domains in $ℂ^{2}$

Jim Arlebrink (1993)

Annales de l'institut Fourier

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Let $D$ be a bounded strictly pseudoconvex domain in $ℂ^{2}$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$ . This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in $ℂ^{2}$ .

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. ...