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Vassili Nestoridis

Publications and source records attributed to Vassili Nestoridis.

16 recordsLinked to original sources
arXiv2025

A logarithmic characterization of Arakelian sets

Arakelian's classical approximation theorem \cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\subset \mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^\circ$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.

math.CV↗
arXiv2021

Generic versions of a result in the theory of Hardy spaces

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.

math.CV↗
arXiv2021

Frequently dense harmonic functions and universal martingales on trees

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 $P$) such that, for every measurable function $h$ on the boundary, it contains a subsequence that converges to $h$ in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.

math.FA↗
arXiv2020

Universal Taylor Series in several variables depending on parameters

We establish generic existence of Universal Taylor Series on products $Ω= \prod Ω_i$ of planar simply connected domains $Ω_i$ where the universal approximation holds on products $K$ of planar compact sets with connected complements provided $K \cap Ω= \emptyset$. These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that $K \cap \overlineΩ = \emptyset$ and $\{\infty\} \cup [\mathbb{C} \setminus \overlineΩ_i]$ is connected for all $i$. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper. These universalities are topologically and algebraically generic.

math.CV↗
arXiv2020

Approximation on compact sets of functions and all derivatives

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.

math.CV↗
arXiv2019

An extension of the universal power series of Seleznev

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 be continuous and such that for every complex numbers a_0, a_1, ... , a_{n - 1}, c there exists a complex number a_n such that b_n(a_0, a_1,..., a_{n-1}, a_n) = c. This clearly covers the case of linear functions b_n.

math.CV↗
arXiv2017

A remark on projective limits of function spaces

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, . . . , under the restriction maps is homeomorphic and linearly isomorphic to the Frechet space H(Omega), idependently of the choice of the spaces Xm(Omega_m).

math.CV↗
arXiv2016

Spherical arc-length as a global conformal parameter for analytic curves in the Riemann sphere

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-space and we discuss the situation of curves in the Riemann sphere.

math.CV↗
arXiv2016

Dirichlet approximation and universal Dirichlet series

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.

math.CV↗
arXiv2015

Arc length as a conformal parameter for locally analytic curves

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 curve $γ$(s), the limit sets of the largest extensions, that is, the limit set of the curve as s converges to the end point of the interval.

math.CV↗
arXiv2014

Generic properties of Padé approximants and Padé universal series

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 exhibiting dense or closed infi nite dimensional linear subspaces of analytic functions in a simply connected domain of the complex plane, containing the origin, whose all non zero elements are made of Pad e universal series. We also show how Pad e universal series can be built from classical universal series with large Ostrowski-gaps.

math.FA↗
arXiv2013

Universal Taylor Series On Convex Subsets Of $\Mathbb{C}^{N}$

We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm Ω}\subset {\mathbb C}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact set disjoint from $\bar{\rm Ω}$ and on denumerably many convex compact sets in ${\mathbb C}^n\backslash {\rm Ω}$ which may meet the boundary $\partial {\rm Ω}$. If the universal approximation is only required on convex compact sets disjoint from $\bar{\rm Ω}$, then $f$ may be chosen to be smooth on $\partial {\rm Ω}$, that is $f\in A^{\infty}({\rm Ω})$. Those are generic universalities.

math.CV↗
arXiv2012

Padé Approximants, density of rational functions in $\bbb{A^\infty(\OO)}$ and smoothness of the integration operator

First we establish some generic universalities for Padé 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, even if the boundary $\partial\OO$ of a Jordan domain $\OO$ has infinite length, the integration operator on $\OO$ preserves $H^\infty(\OO)$ and $A(\OO)$ as well.\ We also give an example of a Jordan domain $\OO$ and a function $f\in A(\OO)$, such that its antiderivative is not bounded on $\OO$.\ Finally we restate these results for Volterra operators on the open unit disc $D$ and we complete them by some generic results.

math.CV↗
arXiv2011

Universal Padé Approximation

In transferring some results from universal Taylor series to the case of Padé 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.

math.CV↗