On random frequent universality
type: Item
instance of: scholarly article
title: On random frequent universality
zbMATH Open document ID: 1326.47007
author: Augustin Mouze
author: Vincent Munnier
published in: Journal of Mathematical Analysis and Applications
publication date: 2015-03-27T00:00:00Z
review text: Extending a recent result by \textit{M. Nikula} [Complex Anal. Oper. Theory 8, No. 7, 1455--1474 (2014; Zbl 1315.47010)], the authors provide probabilistic proofs of known results about frequent hypercyclicity and universality. Namely, by an application of Kolmogorov's Zero-One Law and Birkhoff's Ergodic Theorem, it is shown that for a sequence of complex iid random variables \((X_n)_{n\in\mathbb{N}}\) whose distribution is supported by the whole plane \(\mathbb{C}\) and satisfies a certain decay property at infinity, the random power series \(f(z)=\sum_{n\geq 0}X_n\frac{z^n}{n!}\) is almost surely frequently hypercyclic for the differentiation operator on the Fréchet space of entire functions \(H(\mathbb{C})\) and has almost surely a frequently universal generalized Taylor expansion at the origin. Furthermore, the authors use their arguments to derive some new results about frequent hypercyclicity for certain polynomials of weighted backward shifts on \(\ell_p\).
reviewed by: Thomas Kalmes
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 30B20
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 6420114
zbMATH Keywords: frequently hypercyclic operator
zbMATH Keywords: frequently universal series
zbMATH Keywords: random power series
zbMATH Keywords: ergodic theory
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.11.005
OpenAlex ID: W2091197551
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DOI: 10.1016/J.JMAA.2013.11.005
description: scientific article; zbMATH DE number 6420114