Approximate Carleman theorems and a Denjoy-Carleman maximum principle
type: Item
instance of: scholarly article
title: Approximate Carleman theorems and a Denjoy-Carleman maximum principle
zbMATH Open document ID: 1056.26017
author: Isabelle Chalendar
author: Laurent Habsieger
author: Thomas J. Ransford
author: Jonathan R. Partington
published in: Archiv der Mathematik
publication date: 2004-09-27T00:00:00Z
full work available at URL: partingtonjr11 dcpaper.pdf
review text: Let \((M_n)_{n\geq 0}\) be a positive sequence satisfying \(M_0=1\), \(M_n^2\leq M_{n-1}M_{n+1}\) \((n\geq 1)\) and \(\sum_{n=1}^{\infty}M_n^{-1/n}=\infty.\) The authors show that if \(f\) is in the Carleman class \(C_{\mathbb R}(M_n):=\{f\in C^\infty(\mathbb R): |f^{(n)}(x)| \leq c_f\rho_f^nM_n\), \(x\in\mathbb R\), \(n\geq 0\}\) and \(\limsup| f^{(n)}(0)| ^{1/n}\leq C\), then for all integers \(m,n\geq 0\) \[ \sup_{x\in\mathbb R}| f(n+m)(x)| \leq C^n\sup_{x\in\mathbb R}| f^{(m)}(x)| . \] If, in particular, \(C=0\) then one obtains a generalization of the well-known Denjoy-Carleman theorem (\(f^{(n)}=0\) for \(n=0,1,\dots \implies f(x)=0,\;x\in\mathbb R\)). As an application of the above ideas, the authors obtain a generalization of Carleman's theorem on the unique determination of probability measures by their moments.
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 26E10
Mathematics Subject Classification ID: 44A60
zbMATH DE Number: 2103669
zbMATH Keywords: Carleman classes
zbMATH Keywords: Denjoy-Carleman maximum principle
Wikidata QID: Q56319352
MaRDI profile type: Publication
OpenAlex ID: W1979417373
DOI: 10.1007/S00013-004-0608-Z
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description: scientific article; zbMATH DE number 2103669