Vasilis Nestoridis

Vasilis Nestoridis
National and Kapodistrian University of Athens | uoa · Department of Mathematics and Informatics

About

129
Publications
4,622
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1,213
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Additional affiliations
September 1997 - present
National and Kapodistrian University of Athens
Position
  • Professor (Full)
September 1985 - September 1997
University of Crete
Position
  • Professor (Assistant)

Publications

Publications (129)
Article
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Given a pair of topological vector spaces X,Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X,\; Y$$\end{document} where X is a proper linear subspace of Y it is exami...
Article
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We establish generic existence of Universal Taylor Series on products $$\varOmega = \prod \varOmega _i$$ Ω = ∏ Ω i of planar simply connected domains $$\varOmega _i$$ Ω i where the universal approximation holds on products K of planar compact sets with connected complements provided $$K \cap \varOmega = \emptyset $$ K ∩ Ω = ∅ . These classes are wi...
Article
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In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.
Preprint
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We examine topological and algebraic genericity and spaceability for any pair $(X,Y)$, $X\subset Y$, $X\neq Y$ belonging to an extended chain of sequence spaces which contains the $\ell^p$ spaces, $0<p\leq \infty$.
Article
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In this paper, we show that one-sided extendability of functions in certain \(L^{\infty }\) spaces of a rectifiable Jordan arc is a rare phenomenon. We also discuss possible generalizations.
Article
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Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence of universal Taylor series on products of planar simply connected domains \({\varOmega }_i\), \(i=1,\ldots ,d\). The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar c...
Preprint
Full-text available
For 0<p<1 and f a function in the Hardy space of order p its primitive belongs to the Hardy space q=p/1-p. We show that generically the primitive does not belong, even not locally, in any Hardy space smaller than the Hardy space of order q.
Article
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The celebrated theorem of Mergelyan states that, if K is a compact subset of the complex plane with connected complement, then every continuous function on K which is holomorphic on its interior can be uniformly approximated on K by polynomials. This paper is concerned with polynomial and rational approximation in several complex variables, where t...
Article
Recently, harmonic functions and frequently universal harmonic functions on a tree T have been studied, taking values on a separable Fréchet space E over the field C or R. In the present paper, we allow the functions to take values in a vector space E over a rather general field F. The metric of the separable topological vector space E is translati...
Article
On a large class of infinite trees T T , we prove the existence of harmonic functions h h , with respect to suitable transient transition operators P P , that satisfy the following universal property: h h is the Poisson transform of a martingale on the end-point boundary Ω \Omega of T T (equipped with the harmonic measure induced by P P ) such that...
Article
We prove concurrent universal Padé approximation for several universal Padé approximants of several types. Our results are generic in the space of holomorphic functions, in the space of formal power series as well as in a subspace of A∞. These results are valid for one center of expansion or for several centers as well. The Padé approximants allow...
Article
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We show generic existence of power series \(a = \sum \nolimits _{n = 0}^{\infty }a_nz^n, a_n \in \mathbb {C}\), such that the sequence \(T_N(a)(z) = \sum \nolimits _{n=0}^{N}b_n(a_0, \ldots , a_n)z^n, N = 0, 1, 2 \ldots \) approximates every polynomial uniformly on every compact set \(K \subset \mathbb {C}{\setminus }\{0\}\) with connected compleme...
Preprint
Recently, harmonic functions and frequently universal harmonic functions on a tree $T$ have been studied, taking values on a separable Fr\'{e}chet space $E$ over the field $\mathbb{C}$ or $\mathbb{R}$. In the present paper, we allow the functions to take values in a vector space $E$ over a rather general field $\mathbb{F}$. The metric of the separa...
Preprint
We establish generic existence of Universal Taylor Series on products $\Omega = \prod \Omega_i$ of planar simply connected domains $\Omega_i$ where the universal approximation holds on products $K$ of planar compact sets with connected complements provided $K \cap \Omega = \emptyset$. These classes are with respect to one or several centers of expa...
Preprint
In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.
Preprint
We present the example of l^p spaces, where we examine results of topological and algebraic genericity and spaceability. At the end of the paper we include a project with other chains of spaces, mainly of holomorphic functions, where a similar investigation can be done.
Article
We show that the set of frequently universal harmonic functions on a tree T contains a vector space except 0 which is dense in the space of harmonic functions on T seen as subset of CT. In order to prove this we replace the complex plane C by any separable Fréchet space E and we repeat all the theory.
Preprint
Using a recent Mergelyan type theorem for products of planar compact sets we establish generic existence of Universal Taylor Series on products of planar simply connected domains Omegai, i=1, . . . , d. The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets Ki,...
Preprint
We show that the set of frequently universal harmonic functions on a tree T contains a vector space except 0 which is dense in the space of harmonic functions on T seen as subset of C^T . In order to prove this we replace the complex plane C by any separable Frechet space E and we repeat all the theory.
Preprint
Full-text available
We prove the existence of harmonic functions f on trees, with respect to suitable transient transition operators P, that satisfy an analogue of Menshov universal property in the following sense: f is the Poisson transform of a martingale on the boundary of the tree (equipped with the harmonic measure m induced by P) that, for every measurable funct...
Article
We show that if an open arc J of the boundary of a Jordan domain \({\varOmega }\) is rectifiable, then the derivative \(\varPhi '\) of the Riemann map \(\varPhi :D\;\rightarrow \;{\varOmega }\) from the open unit disk D onto \({\varOmega }\) behaves as an \(H^1\) function when we approach the arc \(\varPhi ^{-1}(J')\), where \(J'\) is any compact s...
Article
For a function space X(Ω) satisfying weak assumptions we prove that the generic function in X(Ω) is totally unbounded, hence non-extendable. We provide several examples of such spaces; they are mainly localized versions of classical function spaces and intersections of them.
Article
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It is well known that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. If \(X({\varOmega })\) is a space of holomorphic functions we extend these notions to \(X({\varOmega })\)-domain of holomorphy and weak \(X({\varOmega })\)-domain of holomorphy. For several function spaces \(X(\varOmega )\), satisfying weak assum...
Preprint
We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on every compact set K not containing the origin and with connected complement. The functions b_n are assumed to...
Preprint
For compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
Preprint
For a function space $X(\OO)$ satisfying weak assumptions we prove that the generic function in $X(\OO)$ is totally unbounded, hence non-extendable. We provide several examples of such spaces; they are mainly localized versions of classical function spaces and intersections of them.
Article
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Let \(\varOmega \) be a Jordan domain in \(\mathbb {C}\), J an open arc of \(\partial \varOmega \) and \(\phi : D\rightarrow \varOmega \) a Riemann map from the open unit disk D onto \(\varOmega \). Under certain assumptions on \(\phi \) we prove that if a holomorphic function \(f\in H(\varOmega )\) extends continuously on \(\varOmega \cup J\) and...
Article
In this paper I present my first proof regarding the existence of universal Taylor series on the disc where the universal approximation was required on the boundary as well. It is a modification of a construction giving a negative answer to a question of S. Pichorides, where the approximation was valid only on the boundary of the disc. There was no...
Article
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We provide with a simple and explicit construction of a function in the disc algebra, whose derivatives enjoy a disjoint universal property near the boundary. The set of functions with such property is topologically generic, densely lineable, and spaceable.
Chapter
We construct families of universal Taylor series on Ω depending on a parameter w ∈ G, where Ω and G are planar simply connected domains. The functions to be approximated depend on the parameter w, w ∈ G. The partial sums implementing the universal approximation are one variable partial sums with respect to z ∈ Ω for each fixed value of the paramete...
Chapter
We present a sufficient condition to ensure the density of the set of rational functions with prescribed poles in the algebra A∞ (Ω).
Article
Let Ω be a Jordan domain in C, J an open arc of ∂Ω and φ : D → Ω a Riemann map from the open unit disk D onto Ω. Under certain assumptions on φ we prove that if a holomorphic function f ∈ H(Ω) extends continuously on Ω ∪ J and p ∈ {1, 2, . . . } ∪ {∞}, then the following equivalence holds: the derivatives f (l) , 1 ≤ l ≤ p, l ∈ N, extend continuous...
Article
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We consider generalizations of classical function spaces by requiring that a holomorphic in {\Omega} function satisfies some property when we approach from {\Omega}, not the whole boundary, but only a part of it. These spaces endowed with their natural topology are Fr\'echet spaces. We prove some generic non-extendability results in such spaces and...
Article
Let X be a normed linear space. We examine if every open, convex and unbounded subset of X is equal to the union of a family of open straight half lines. The answer is affirmative if and only if X is finite dimensional.
Article
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We consider the spaces $H_{F}^{\infty}(\Omega)$ and $\mathcal{A}_{F}(\Omega)$ containing all holomorphic functions $f$ on an open set $\Omega \subseteq \mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F \subseteq \mathbb{N}_0=\{ 0,1,...\}$, are bounded on $\Omega$, or continuously extendable on $\overline{\Omega}$, respectively. We endow the...
Article
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We show that if an open arc J of the boundary of a Jordan domain $\Omega$ is rectifiable, then the derivative $\Phi$' of the Riemann map $\Phi: D\rightarrow \Omega$ from the open unit disk D onto $\Omega$ behaves as an $H^1$ function when we approach the arc $\Phi^{-1}(J^{\prime})$,where $J^{\prime}$ is any compact subarc of $J$. "
Article
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Let Omega subset of C^d be an open set and Km, m = 1, 2, . . . an exhaustion of Omega by compact subsets of Omega. We set Omega_m = int(Km) and let Xm(Omega_m) be a topological space of holomorphic functions on Omega_m between A^ infinity (Omega_m) and H(Omega_m). Then we show that the projective limit of the family Xm(Omega_m), m = 1, 2, . . . , u...
Article
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We consider Bergman spaces and variations of them in one or several complex variables. For some domains we show that in these spaces the generic function is totally unbounded and hence non - extendable. We also show that the generic function is not - even locally - in Bergman spaces of higher order. Finally, in certain domains we consider the space...
Article
We give a simple and more elementary proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire's Category Theorey and Montel's Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of holomorphic functions i...
Article
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If a Jordan curve {\sigma} has a one-sided conformal collar with "good" properties, then, using the Reflection principle, we show that any other conformal collar of {\sigma} from the same side has the same "good" properties. A particular use of this fact concerns analytic Jordan curves, but in general the Jordan arcs we consider do not have to be a...
Article
We consider \(A(\Omega )\), the Banach algebra of all functions f from \(\overline{\Omega }=\prod _{i\in I}\overline{U_{i}}\) to \( \mathbb C\) that are continuous on \(\overline{\Omega }\) with respect to the product topology and separately holomorphic in \(\Omega \), where I is an arbitrary set and \(U_{i}\) are planar domains of some type. We sh...
Article
We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Euclidean n-space and complex n-s...
Article
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In the present note we examine possible extensions of Runge, Mergelyan and Arakelian Theorems, when the uniform approximation is meant with respect to the metric ρ of a metrizable compactification (S,ρ) of the complex plane.
Article
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For a non-simply connected domain Omega in C and f a holomorphic function on Omega we prove that f admits one-valued primitives of any order in Omega, if and only if, it extends holomorphically in the simply connected envelope of Omega . This leads to a generalization of the F. and M. Riesz theorem.
Article
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We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical results of Runge, Mergelyan and Vitushkin. We also strengthen the notion of universal Dirichlet series.
Article
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Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to introduce the p-continuous analytic capacity and variants of it, $p \in \{ 0, 1, 2, \cdots \} \cup \{ \infty \}$, for...
Article
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We consider an infinite locally finite tree T equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on T has dense range in . By looking at forward-only transition coefficients, we show that the generic harmonic function induces a b...
Article
We show that arc length is a global conformal parameter for analytic curves and that this parameter can be used to decide whether the domain of definition of an analytic curve can be extended or not. The maximal extension with respect to the arc length parameter is the largest possible extension (over all parametrizations of the curve). Our proof i...
Article
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In this article we show that extendability from one side of a simple analytic curve is a rare phenomenon in the topological sense in various spaces of functions. Our result can be proven using Fourier methods combined with other facts or by complex analytic methods and a comparison of the two methods is possible.
Article
We establish properties concerning the distribution of poles of Padé approximants, which are generic in Baire category sense. We also investigate Padé universal series, an analog of classical universal series, where Taylor partial sums are replaced with Padé approximants. In particular, we complement previous studies on this subject by exhibiting d...
Article
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For any locally analytic curve we show that arc length can be complexified and seen as a conformal parameter. As an application, we show that any such curve defines a unique maximal one and that the notions of analytic Jordan curve coincides with the notion of a Jordan curve which is locally analytic. We give examples where we also find, for a curv...
Article
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We prove simultaneous universal Pad\'{e} approximation for several universal Pad\'{e} approximants of several types. Our results are generic in the space of holomorphic functions, in the space of formal power series as well as in a subspace of $A^{\infty}$. These results are valid for one center of expansion or for several centers as well.
Article
First we establish some generic universalities for Padé approximants in the closure in the space of all rational functions with poles off . The closure of the domain is taken with respect to the finite plane . Next we give sufficient conditions on Ω so that . Some of these conditions imply that, even if the boundary ∂Ω of a Jordan domain Ω has infi...
Article
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The theory of universal Taylor series can be extended to the case of Pad\'e approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the Pad\'e approximants of some power series. We present the first generic result in this direction, for Pad\'e approximants corresponding to Taylor dev...
Article
We establish the existence of universal Laurent series on some domains of infinite connectivity. This phenomenon is topologically and algebraically generic.
Article
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We identify the complex plane [Inline formula] with the open unit disc [Inline formula] by the homeomorphism [Inline formula]. This leads to a compactification [Inline formula] of [Inline formula], homeomorphic to [Inline formula]. The Euclidean metric on [Inline formula] induces a metric [Inline formula] on [Inline formula]. We identify all unifor...
Article
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For a function defined on an arbitrary subset of a Riemann surface, we give conditions which allow the function to be extended conformally. One folkloric consequence is that two common definitions of an analytic arc in ${\mathbb C}$ are equivalent. Key-words: analytic continuation; analytic arc.
Article
Full-text available
We establish properties concerning the distribution of poles of Pad e approximants, which are generic in Baire category sense. We also investigate Pad e universal series, an analog of classical universal series, where Taylor partial sums are replaced with Pad e approximants. In particular, we complement previous studies on this subject by exhibitin...
Article
We give sufficient conditions on planar domains for polynomials to be dense in the algebras A and A-infinity of the product of these domains, endowed with their natural topologies. We also characterize the uniform limits, with respect to the chordal metric, of polynomials on the product of the closures of these domains. The products may be finite p...
Article
We show that universal Taylor series in unbounded non-simply connected domains can be represented as series of rational functions with a double simultaneous approximation property. The use of Baire’s category theorem allows us to obtain strong results. Moreover, we extend our results from the holomorphic case to the meromorphic one, where we use th...
Article
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We extend Mergelyan’s theorem to the case of compact sets K of the plane, bounded by a finite number of disjoint Jordan curves, where the approximation is uniform with respect to the chordal metric χ and it is realized by rational functions with prescribed poles off K. Allowing poles in the interior of K, as well, we obtain an analogous result with...
Article
We use the chordal metric in order to approximate all meromorphic functions on ${\mathbb{C} \backslash \{0\}}$ by Padé approximants of formal power series. This is a generic universality of Seleznev type which implies Menchov type almost everywhere approximation with respect to any σ-finite Borel measure on ${\mathbb{C} \backslash \{0\}}$ .
Article
We investigate the sets of uniform limits $A(\bar{B}_n)$, $A(\bar{D}^I)$ of polynomials on the closed unit ball $\bar{B}_n$ of $\mathbb{C}^n$ and on the cartesian product $\bar{D}^I$ where $I$ is an arbitrary set and $\bar{D}$ is the closed unit disc in $\mathbb{C}$. We introduce the notion of set of uniqueness for $A(\bar{D}^I)$ (respectively for...
Article
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We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm \Omega}\subset {{\mathbb C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from $\bar{{\rm \Omega}}$ and on denumerably many convex compact sets in ${{\mathbb C}}^n\backslash...
Article
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First we establish some generic universalities for Pad\'{e} approximants in the closure $X^\infty(\OO)$ in $A^\infty(\OO)$ of all rational functions with poles off $\oO$, the closure taken in $\C$ of the domain $\OO\subset\C$.\ Next we give sufficient conditions on $\OO$ so that $X^\infty(\OO)=A^\infty(\OO)$.\ Some of these conditions imply that, e...
Article
Let Ω be an arbitrary domain in the complex plane, Ω ≠ ℂ, and ζ ∈ Ω. Let R = dist(ζ, ∂Ω) Ω (0, +∞), C(ζ, R) = {z ∈ ℂ: |z − ζ| = R} and J(Ω, ζ) = ∂Ω ∩ C(ζ, R). If f is a holomorphic function in Ω, then its Taylor series with center at Ω, Σ n=0∞ c n (z − ζ)n , universal with respect to J(Ω, ζ), if the sequence of its partial sums approximates uniform...
Article
A domain Ω is called a domain of injective holomorphy if there exists an injective holomorphic function f : Ω → C that is non-extendable. We give examples of domains that are domains of injective holomorphy and others that are not. In particular, every regular domain (Ω = Ω) is a domain of injective holomorphy, and every simply connected domain is...
Article
The use of the chordal distance does not change the notion of Universal Taylor series. However, it changes the notion of Universal Padé approximants. Using Padé approximants of meromorphic or holomorphic functions we can approximate all rational functions on compact sets of arbitrary connectivity.
Article
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The known proofs for universal Taylor series do not determine a specific universal Taylor series. In the present paper, we isolate a specific universal Taylor series by modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific universal Taylor series on the disc or on a polygonal domain. Furthermore in non simply connecte...
Article
We introduce a new class of Universal series, in the abstract setting. This class lies between two already known classes of Universal series. By two examples we differentiate these classes. These examples are the first examples in the case where all classes are non-void. We also prove that the class of Universal series where the approximating index...
Data
Résumé A toute compactification metrizable S du plan complexe, nous associons une extension A(D, S) de l' algebre du disc A(D). Un cas fondamental est le cas S = C ∪ {∞}. Nous déterminons l' ensemble de limites uniformes des polynômes sur le disc unité fermé D, par rappor a la metrique chordale ; ensuite nou etendons cetté etude dans le cas géneral...
Article
Full-text available
The known proofs for universal Taylor series do not determine a specific universal Taylor series. In the present paper, we isolate a specific universal Taylor series by modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific universal Taylor series on the disc or on a polygonal domain. Furthermore in non simply connecte...
Article
Full-text available
We prove the existence of series ∑anψn, whose coefficients (an) are in ∩p>1ℓp and whose terms (ψn) are translates by rational vectors in Rd of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener’s algebra W(C0,ℓ1), Cb(Rd), C0(Rd), Lp(Rd), for every p∈[1,∞), an...
Article
Inspired by a lemma of Zalcman, we show that certain universal sequences of holomorphic functions can always be chosen to be non-normal. This leads to a new vision of universality in connection with non-normality.
Article
We investigate the uniform limits of the set of polynomials on the closed unit disc D¯ with respect to the chordal metric χ. More generally, we examine analogous questions replacing C∪{∞} by other metrizable compactifications of C.
Article
Generic approximation of entire functions by their Pad\'{e} approximants has been achieved in the past (\cite{3}). In the present article we obtain generic approximation of holomorphic functions on arbitrary open sets by sequences of their Pad\'{e} approximants. Similar results hold with functions smooth on the boundary of their domain of definitio...
Article
Full-text available
We investigate the set of uniform limits of polynomials on any closed Jordan domain with respect to the chordal metric $\chi$ on $\mathbb{C}\cup\{\infty \}$. We conclude that Mergelyan's Theorem may be extended to the case of uniform approximation with respect to $\chi$ on closed Jordan domains. Similar results are obtained if we replace the one po...
Article
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In transferring some results from universal Taylor series to the case of Pad\'e approximants we obtain stronger results, such as, universal approximation on compact sets of arbitrary connectivity and generic results on planar domains of any connectivity and not just on simply onnected domains.
Article
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We identify the complex plane C with the open unit disc D={z:|z|<1} by the homeomorphism z --> z/(1+|z|). This leads to a compactification $\bar{C}$ of C, homeomorphic to the closed unit disc. The Euclidean metric on the closed unit disc induces a metric d on $\bar{C}$. We identify all uniform limits of polynomials on $\bar{D}$ with respect to the...
Article
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The purpose of this article is to establish extensions of Fekete's Theorem concerning the existence of universal power series of C ∞ functions defined by estimates on successive derivatives.
Article
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We identify all uniform limits of polynomials on the closed unit disc D̅ with respect to the chordal metric χ on ℂ∪{∞}. One such limit is f≡∞. The other limits are holomorphic functions f : D→ℂ so that, for every ζ∈∂ D, the exists in ℂ∪{∞}. The above class of functions is an extension of the disc algebra and is denoted by Ã(D). We study properties...
Article
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In this paper an abstract condition is given yielding universal series defined by sequences a = {aj}∞j=1 in ∩p>1ℓp but not in ℓ1. We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann ζ-function, or a fundamental solution of a given elliptic operator in ℝν with constant coef...
Article
Let Ω⊂CΩ⊂C, Ω≠CΩ≠C be any domain and ζ∈Ωζ∈Ω. Let R=dist(ζ,Ωc)∈(0,+∞)R=dist(ζ,Ωc)∈(0,+∞) and C(ζ,R)={z∈C:|ζ−z|=R}. We set J(Ω,ζ)=Ωc∩C(ζ,R)J(Ω,ζ)=Ωc∩C(ζ,R). Then there exists f∈H(Ω)f∈H(Ω), such that the sequence SN(f,ζ)(z)=∑n=0Nf(n)(ζ)n!(z−ζ)n, N=0,1,…N=0,1,… , approximates any polynomial uniformly on each compact set K⊂J(Ω,ζ)K⊂J(Ω,ζ) with C∖KC∖K con...
Article
We prove that a function f holomorphic in a simply connected domain Omega whose Taylor series at xi is an element of Omega is universal with respect to overconvergence automatically has a strong kind of universality: its expansion in Faber series corresponding to any connected compact set Gamma subset of Omega with (C) over cap\Gamma connected is u...
Article
Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence of scalars (α j) ∞ j=1 , there exists a subsequence (α kj) ∞ j=1 , such that either every subsequence of (α kj) ∞ j=1 defines a universal series, or every subsequence of (α kj) ∞ j=1 does not define a universal series....
Article
We give an abstract framework for the theory of universal series, from which we deduce easily and in a unified way most of the existing results as well as new and stronger statements.
Article
Let › be a Jordan region in C and let f 2 H(›) be a universal function (in various senses). In this paper, we prove that it is possible for f to have simulta- neously the following properties: (i) f is univalent, (ii) each derivative f(') (' ‚ 0) extends continuously on the closure of ›.
Article
We prove that universal approximation (uniform approximation on compact subsets with connected complement) implies almost everywhere approximation in the sense of Menchoff with respect to any given σ-finite Borel measure on (d≥2).
Article
In the case of the complex plane, several notions of universal Taylor series have been introduced. The purpose of the present article is to establish the existence of universal Taylor series of C ∞ functions in general open subsets of the Euclidean space ℝ n , n≥1.
Article
Let Ω be a simply connected domain in ℂ For a function f holomorphic in Ω let S n (f, ζ) denote the partial sum of the Taylor development of f with center ζ ∈ Ω. We show that generically overconvergence phenomena of S n(f, ζ) and their derivatives can occur on the boundary ∂Ω or in parts of it. In the rest of the boundary ∂Ω, the universal function...
Article
Let W Í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {¥} È[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and z Î [`(W)]\zeta...
Article
We present an abstract theory of universal series; in particular, we give a necessary and sufficient condition for the existence of universal series of a certain type. Most of the known results can be proved or strengthened by using this condition. We also obtain new results, for example, related to universal Dirichlet series. To cite this article:...
Article
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Universal Taylor series are defined on simply connected domains, but they do not exist on an annulus. Instead we introduce universal Laurent or Laurent–Faber series on finitely connected domains in $\mathbb{C}$. These are generic universalities. Furthermore, we study some properties of universal Laurent series on an annulus.
Article
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We modify arguments used in the study of Universal Taylor Series and extend them from simply connected domains to general domains in $\bb {C}$ to give a new simple and natural proof of the fact that all holomorphic functions are generically non extendable.

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