Interpolation by hypercyclic functions for differential operators
type: Item
instance of: scholarly article
title: Interpolation by hypercyclic functions for differential operators
zbMATH Open document ID: 1182.41002
published in: Journal of Approximation Theory
publication date: 2009-05-28T00:00:00Z
review text: In 1929, \textit{G. D. Birkhoff} [C. R. Acad. Sci. Paris 189, 473--475 (1929; JFM 55.0192.07)] proved the existence of an entire function \(f\) which is universal in the sense that the sequence \(\{f(\cdot +an ):n\geq 1\}\) of its translates (\(a\in \mathbb{C} \backslash \{0\}\)) is dense in the space \(H(\mathbb{C})\) of all entire functions. Later, in 1952, \textit{G. R. MacLane} [J. Anal. Math. 2, 72--87 (1952; Zbl 0049.05603)] demonstrated the same density property for the sequence \(\{f^{(n)}:n\geq 1\}\) of derivatives of some entire function \(f\). Formally speaking, the translation operator \(\tau _{\alpha }:f\mapsto f(\cdot +a)\) and the derivation operator \(D:f\mapsto f'\) acting on the space \(H(\mathbb{C})\) are hypercyclic. In 1991, \textit{G. Godefroy} and \textit{J. H. Shapiro} [J. Funct. Anal. 98, No. 2, 229--269 (1991; Zbl 0732.47016)] unified both the above results by establishing the hypercyclicity on \(H(\mathbb{C})\) of any nonscalar differential operator \(\varphi (D)\), generated by an entire function \(\varphi \) of exponential type. Birkhoff's theorem and MacLane's theorem correspond to the cases \(\varphi (z)=\exp(az)\), \(\varphi (z)=z\), respectively. In a different direction, the Weierstrass interpolation theorem asserts the existence of holomorphic functions on a domain \(\Omega\) of the complex plane \(\mathbb{C}\) having prescribed values at a sequence of points without accumulation points. Recently, Costakis and Vlachou in a preprint with apparently no further references, and independently \textit{M. Niess} [Analysis, München 27, No. 2--3, 323--332 (2007; Zbl 1132.30303), Complex Var. Elliptic Equ. 53, No. 9, 819--831 (2008; Zbl 1153.30003)] carried out the interesting task of combining both properties of hypercyclicity and interpolation. They proved the existence of a holomorphic function \(f\) that is MacLane-universal and simultaneously takes prescribed values at the points of a given set without accumulation points in \(\Omega\),…
reviewed by: Daniel Cárdenas Morales
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 5558975
zbMATH Keywords: hypercyclic function
zbMATH Keywords: differential operator
zbMATH Keywords: interpolation
zbMATH Keywords: mixing sequence of mappings
author: L. Bernal-González
MaRDI profile type: Publication
full work available at URL: j.jat.2008.07.003
OpenAlex ID: W2083758308
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DOI: 10.1016/J.JAT.2008.07.003
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description: scientific article; zbMATH DE number 5558975