A counterexample in Hardy spaces with an application to systems theory
type: Item
instance of: scholarly article
title: A counterexample in Hardy spaces with an application to systems theory
zbMATH Open document ID: 0841.30031
DOI: 10.4171/ZAA/648
author: V. Katsnelson
author: George Weiss
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1996-03-17T00:00:00Z
review text: Summary: We know from elementary calculus that if \(f\) is a differentiable function on \((0, \infty)\) with \(f(\infty) = 0\), then \(\lim_{x \to + \infty} xf(x) = - \lim_{z \to + \infty} x^2 f'(x)\) in the following sense: if the second limit exists, then so does the first and they are equal. The existence of the first limit does not imply the existence of the second, in general. It was hoped that the existence of the first limit, together with the assumption that \(f\) belongs to the Hardy class \(H^2\), does imply the existence of the second limit. This would have simplified certain results in systems theory. We show by counterexample that this is not the case. More precisely, for each \(q > 1\) (including \(q = \infty)\) we construct a function \(f\) in the Hardy space \(H^q\) on the right half-plane such that \(\lim_{x \to + \infty} xf(x) = 0\) and \(\limsup_{x \to + \infty} x^2 |f'(x) |= \infty\). A special Blaschke product plays a crucial role in this construction.
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 93C25
zbMATH DE Number: 856791
MaRDI profile type: Publication
Wikidata QID: Q124810949
cites work: Q3920176
cites work: Q3872710
cites work: Q5799604
cites work: Q5618011
cites work: Q5516901
cites work: Realization theory in Hilbert space
cites work: Infinite Dimensional Linear Systems With Unbounded Control and Observation: A Functional Analytic Approach
cites work: Admissible observation operators for linear semigroups
cites work: Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity
cites work: Regular linear systems with feedback
full work available at URL: https://doi.org/10.4171/zaa/648
OpenAlex ID: W2042197953
description: scientific article; zbMATH DE number 856791