Finely holomorphic functions and quasi-analytic classes
type: Item
instance of: scholarly article
title: Finely holomorphic functions and quasi-analytic classes
zbMATH Open document ID: 0826.31004
DOI: 10.1007/BF01468247
author: Pavel Pyrih
published in: Potential Analysis
publication date: 1994-10-09T00:00:00Z
review text: Let \(C\{M_n\}\) be the class of all complex functions \(f: \mathbb{R}\to \mathbb{C}\), \(f\in C^\infty\), which satisfy the inequalities \(|f^{(n)} (x)|< \beta_f B_f M_n\) \((n=0,1, 2,\dots)\), where \(f^{(0)} =f\) and \(\beta_f\), \(B_f\) are constants depending on \(f\), while the constants \(M_n\) satisfy the following inequalities: \(M_0 =1\), \(M^2_n\leq M_{n-1} M_{n+1}\) \((n=1,2, \dots)\). A class \(C\{ M_n\}\) is said to be quasi-analytic if \(f^{(n)} (0)=0\) \((n=0, 1,2, \dots)\) imply \(f(x)=0\) \(\forall x\in \mathbb{R}\). Let \({\mathcal F}\) denote the class of complex functions \(f: \mathbb{R}\to \mathbb{C}\), for which there exists a fine domain \(U\) containing the real axis and a function \(\widetilde {f}\) finely holomorphic on \(V\) [cf. \textit{B. Fuglede}, Proc. 18th Scand. Congr. Math., Aarhus 1980, 22-38 (1981; Zbl 0462.30004)] that satisfies \(f(x)= \widetilde {f} (x)\) \(\forall x\in \mathbb{R}\). Let \(Q\) denote the class of complex functions \(f: \mathbb{R}\to \mathbb{C}\) for which there is a quasi-analytic class \(C\{ M_n\}\) containing \(f\). The author establishes that \(Q-{\mathcal F}\) as well as \({\mathcal F}-Q\) are non empty.
Mathematics Subject Classification ID: 31C40
Mathematics Subject Classification ID: 30D60
zbMATH DE Number: 650842
zbMATH Keywords: fine topology
zbMATH Keywords: finely holomorphic functions
zbMATH Keywords: quasi-analytic class
reviewed by: Petru Caraman
MaRDI profile type: Publication
cites work: Sur les fonctions finement holomorphes
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description: scientific article; zbMATH DE number 650842