Some remarks on universal functions and Taylor series

2000 ◽  
Vol 128 (1) ◽  
pp. 157-175 ◽  
Author(s):  
G. COSTAKIS
Keyword(s):  
Complex Plane ◽  
Taylor Series ◽  
Connected Domain ◽  
Constructive Proof ◽  
Universal Functions ◽  
Baire’S Theorem ◽  
Simply Connected Domain ◽  
Universal Taylor Series ◽  
Simply Connected

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

scholarly journals Ideal Convergence ofk-Positive Linear Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Akif Gadjiev ◽  
Oktay Duman ◽  
A. M. Ghorbanalizadeh
Keyword(s):  
Complex Plane ◽  
Connected Domain ◽  
Analytic Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Ideal Convergence ◽  
Simply Connected Domain ◽  
Convergence Results ◽  
Simply Connected

We study some ideal convergence results ofk-positive linear operators defined on an appropriate subspace of the space of all analytic functions on a bounded simply connected domain in the complex plane. We also show that our approximation results with respect to ideal convergence are more general than the classical ones.

Universal Taylor series on non-simply connected domains

Analysis ◽  
2006 ◽  
Vol 26 (3) ◽  
Author(s):  
George Costakis ◽  
Vagia Vlachou
Keyword(s):  
Taylor Series ◽  
Connected Domain ◽  
Holomorphic Functions ◽  
Generic Property ◽  
Universal Taylor Series ◽  
Doubly Connected Domain ◽  
Simply Connected ◽  
Simply Connected Domains

In the present paper, we investigate the existence of universal Taylor series on certain non-simply connected domains. Moreover, we prove that Hadamard-Ostrowski gaps is a generic property in the space of holomorphic functions on a doubly connected domain.

scholarly journals On ideal convergence of sequences of linear operators in the space of analytic functions

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 241-251
Author(s):  
Nursel Çetin
Keyword(s):  
Complex Plane ◽  
Connected Domain ◽  
Analytic Functions ◽  
Linear Operators ◽  
Space Of Analytic Functions ◽  
Ideal Convergence ◽  
Simply Connected Domain ◽  
Convergence Of Sequences ◽  
Simply Connected

We investigate the problem of ideal convergence of the sequences of linear operators without the properties of k-positivity in the space of analytic functions in a bounded simply connected domain of complex plane.

scholarly journals Universal Taylor Series in Several Variables Depending on Parameters

2021 ◽  
Author(s):  
G. Gavrilopoulos ◽  
K. Maronikolakis ◽  
V. Nestoridis
Keyword(s):  
Taylor Series ◽  
Universal Approximation ◽  
Universal Functions ◽  
Compact Sets ◽  
Universal Series ◽  
Several Variables ◽  
Universal Taylor Series ◽  
Generic Existence ◽  
Simply Connected ◽  
Simply Connected Domains

AbstractWe 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 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 $$K \cap \overline{\varOmega } = \emptyset $$ K ∩ Ω ¯ = ∅ and $$\{\infty \} \cup [{\mathbb {C}} {\setminus } \overline{\varOmega }_i]$$ { ∞ } ∪ [ C \ Ω ¯ 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.

scholarly journals The boundary of a simply connected domain

1958 ◽  
Vol 64 (2) ◽  
pp. 45-56 ◽  
Author(s):  
George Piranian
Keyword(s):  
Connected Domain ◽  
Simply Connected Domain ◽  
Simply Connected

scholarly journals Universal Taylor series for non-simply connected domains

2010 ◽  
Vol 348 (9-10) ◽  
pp. 521-524 ◽  
Author(s):  
Stephen J. Gardiner ◽  
Nikolaos Tsirivas
Keyword(s):  
Taylor Series ◽  
Universal Taylor Series ◽  
Simply Connected ◽  
Simply Connected Domains

Metric-geometric relations for a conformal mapping of a simply connected domain onto the exterior of the unit disk

1977 ◽  
Vol 29 (2) ◽  
pp. 111-118 ◽  
Author(s):  
V. V. Andrievskii ◽  
V. I. Belyi
Keyword(s):  
Conformal Mapping ◽  
Unit Disk ◽  
Connected Domain ◽  
Simply Connected Domain ◽  
Simply Connected

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

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