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2018 | Buch

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New Trends in Approximation Theory

In Memory of André Boivin

herausgegeben von: Javad Mashreghi, Ph.D. Myrto Manolaki, Paul Gauthier

Verlag: Springer New York

Buchreihe : Fields Institute Communications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The international conference entitled "New Trends in Approximation Theory" was held at the Fields Institute, in Toronto, from July 25 until July 29, 2016. The conference was fondly dedicated to the memory of our unique friend and colleague, André Boivin, who gave tireless service in Canada until his very last moment of his life in October 2014. The impact of his warm personality and his fine work on Complex Approximation Theory was reflected by the mathematical excellence and the wide research range of the 37 participants. In total there were 27 talks, delivered by well-established mathematicians and young researchers. In particular, 19 invited lectures were delivered by leading experts of the field, from 8 different countries.

The wide variety of presentations composed a mosaic of aspects of approximation theory, highlighting interesting connections with important contemporary areas of Analysis. Primary topics discussed include application of approximation theory (isoperimetric inequalities, construction of entire order-isomorphisms, dynamical sampling); approximation by harmonic and holomorphic functions (especially uniform and tangential approximation), polynomial and rational approximation; zeros of approximants and zero-free approximation; tools used in approximation theory; approximation on complex manifolds, in product domains, and in function spaces; and boundary behaviour and universality properties of Taylor and Dirichlet series.

Inhaltsverzeichnis

Frontmatter

The Life and Work of André Boivin

Abstract

André Boivin will be fondly remembered for many reasons. We shall attempt to convey the impact he has had on the authors of this note (and many others) by describing his wonderful personality and his important contributions in the field of Complex Approximation Theory.

Paul Gauthier, Myrto Manolaki, Javad Mashreghi

A Note on the Density of Rational Functions in A ∞ (Ω)

Abstract

We present a sufficient condition to ensure the density of the set of rational functions with prescribed poles in the algebra A ^∞(Ω).

Javier Falcó, Vassili Nestoridis, Ilias Zadik

Approximation by Entire Functions in the Construction of Order-Isomorphisms and Large Cross-Sections

Abstract

A theorem of Hoischen states that given a positive continuous function \(\varepsilon :\mathbb {R}^t\to \mathbb {R}\), a sequence U ₁ ⊆ U ₂ ⊆… of open sets covering \(\mathbb {R}^t\) and a closed discrete set \(T\subseteq \mathbb {R}^t\), any C ^∞ function \(g:\mathbb {R}^t\to \mathbb {R}\) can be approximated by an entire function f so that for k = 1, 2, …, for all \(x\in \mathbb {R}^t\setminus U_k\) and for each multi-index α such that |α|≤ k,

(a)

|(D ^α f)(x) − (D ^α g)(x)| < ε(x);

(b)

(D ^α f)(x) = (D ^α g)(x) if x ∈ T.

This theorem has been useful in helping to analyze the existence of entire functions restricting to order-isomorphisms of everywhere non-meager subsets of \(\mathbb {R}\), analogous to the Barth-Schneider theorem, which gives entire functions restricting to order-isomorphisms of countable dense sets, and the existence of entire functions f determining cross-sections f ∩ A through everywhere non-meager subsets A of \(\mathbb {R}^{t+1}\cong \mathbb {R}^t\times \mathbb {R}\) whose projection \(\{x\in \mathbb {R}^t:(x,f(x))\in A\}\) onto \(\mathbb {R}^t\) is everywhere non-meager, analogous to the Kuratowski-Ulam theorem which gives for residual sets A in \(\mathbb {R}^{t+1}\), points \(c\in \mathbb {R}\) so that the horizontal section of A determined by c has a residual projection \(\{x\in \mathbb {R}^t:(x,c)\in A\}\) in \(\mathbb {R}^t\). The insights gained from this work have also led to variations on the Hoischen theorem that incorporate the ability to require the values of the derivatives on a countable set to belong to given dense sets or to choose the approximating function so that the graphs of its derivatives cut a small section through a given null set or a given meager set. We discuss these results.

Maxim R. Burke

Approximation by Solutions of Elliptic Equations and Extension of Subharmonic Functions

Abstract

In this review we present the main results jointly obtained by the authors and André Boivin (1955–2014) during the last 20 years. We also recall some important theorems obtained with colleagues and give new applications of the above mentioned results. Several open problems are also formulated.

Paul Gauthier, Petr V. Paramonov

Approximation in the Closed Unit Ball

Abstract

In this expository article, we present a number of classic theorems that serve to identify the closure in the sup-norm of various sets of Blaschke products, inner functions and their quotients, as well as the closure of the convex hulls of these sets. The results presented include theorems of Carathéodory, Fisher, Helson–Sarason, Frostman, Adamjan–Arov–Krein, Douglas–Rudin and Marshall. As an application of some of these ideas, we obtain a simple proof of the Berger–Stampfli spectral mapping theorem for the numerical range of an operator.

Javad Mashreghi, Thomas Ransford

A Thought on Approximation by Bi-Analytic Functions

Abstract

A different approach to the problem of uniform approximations by the module of bi-analytic functions is outlined. This note follows the ideas from Khavinson (On a geometric approach to problems concerning Cauchy integrals and rational approximation. PhD thesis, Brown University, Providence, RI (1983), Proc Am Math Soc 101(3):475–483 (1987), Michigan Math J 34(3):465–473 (1987), Contributions to operator theory and its applications (Mesa, AZ, 1987). Birkhäuser, Basel (1988)), Gamelin and Khavinson (Am Math Mon 96(1):18–30 (1989)) and the more recent paper (Abanov et al. A free boundary problem associated with the isoperimetric inequality. arXiv:1601.03885, 2016 preprint), regarding approximation of \(\overline {z}\) by analytic functions.

Dmitry Khavinson

Chebyshev Polynomials Associated with a System of Continua

Abstract

We establish estimates from above for the uniform norm of the Chebyshev polynomials associated with a system of continua \(K \subset \mathbb {C}\) by constructing monic polynomials with small norms on K. The estimates are exact (up to a constant factor) in the case where K has a piecewise quasiconformal boundary and its complement \(\varOmega =\overline {\mathbb {C}} \setminus K\) has no outward pointing cusps.

Isaac DeFrain

Constrained L2-Approximation by Polynomials on Subsets of the Circle

Abstract

We study best approximation to a given function, in the least square sense on a subset of the unit circle, by polynomials of given degree which are pointwise bounded on the complementary subset. We show that the solution to this problem, as the degree goes large, converges to the solution of a bounded extremal problem for analytic functions which is instrumental in system identification. We provide a numerical example on real data from a hyperfrequency filter.

Laurent Baratchart, Juliette Leblond, Fabien Seyfert

Extremal Bounds of Teichmüller-Wittich-Belinskiı̆ Type for Planar Quasiregular Mappings

Abstract

The theorems of TWB (Teichmüller-Wittich-Belinskiı̆) type imply the local conformality (or weaker properties) of quasiconformal mappings at a prescribed point under assumptions of the finiteness of appropriate integral averages of the quantity K _μ(z) − 1, where K _μ(z) stands for the real dilatation coefficient. We establish the extremal bounds for distortions of the moduli of annuli in terms of integrals in TWB theorems under quasiconformal and quasiregular mappings and illustrate their sharpness by several examples. Some local conditions weaker than the conformality are also discussed.

Anatoly Golberg

Families of Universal Taylor Series Depending on a Parameter

Abstract

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 parameter w ∈ G. The universal approximation extends to mixed partial derivatives. This phenomenon is generic in H( Ω × G).

Evgeny Abakumov, Jürgen Müller, Vassili Nestoridis

Interpolation by Bounded Analytic Functions and Related Questions

Abstract

The paper investigates some interpolation questions related to the Khinchine–Ostrowski theorem, Zalcman’s theorem on bounded approximation, and Rubel’s problem on bounded analytic functions.

Arthur A. Danielyan

On Two Interpolation Formulas for Complex Polynomials

Abstract

We discuss, from various points of view (for example the unicity of nodes), two recent interpolation formulas for algebraic polynomials leading to various Bernstein-Markov type inequalities. We also show that each formula contains, as a special case, the Marcel Riesz interpolation formula for trigonometric polynomials.

Richard Fournier, Stephan Ruscheweyh

Operators with Simple Orbital Behavior

Abstract

In this paper we consider two similarity-invariant classes of operators on a complex Hilbert space. A complete description, in terms of properties of various parts of the spectrum, is obtained for the operators in the closure and for the operators in the interior of each of these classes.

Gabriel T. Prǎjiturǎ

Taylor Series, Universality and Potential Theory

Abstract

Universal approximation properties of Taylor series have been intensively studied over the past 20 years. This article highlights the role that potential theory has played in such investigations. It also briefly discusses potential theoretic aspects of universal Laurent series, universal Dirichlet series, and universal polynomial expansions of harmonic functions.

Stephen J. Gardiner

Subharmonic Images of a Convergent Sequence

Abstract

In this paper we characterize the sequences of possible values of a subharmonic function along a convergent sequence of points. We also discuss some related open questions and possible generalizations.

Paul Gauthier, Myrto Manolaki

Titel: New Trends in Approximation Theory
herausgegeben von: Javad Mashreghi
Ph.D. Myrto Manolaki
Paul Gauthier
Verlag: Springer New York
Electronic ISBN: 978-1-4939-7543-3
Print ISBN: 978-1-4939-7542-6
DOI: https://doi.org/10.1007/978-1-4939-7543-3