Pointwise conditions for analyticity and polynomiality of infinitely differentiable functions
type: Item
instance of: scholarly article
title: Pointwise conditions for analyticity and polynomiality of infinitely differentiable functions
zbMATH Open document ID: 0684.26009
DOI: 10.1016/0022-247X(89)90065-6
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: In this interesting paper the authors pose some problems that arise from the following classical results: Theorem 1 (R. P. Boas jun., 1935). Suppose \(f\in C^{\infty}(I)\) where I is a real interval. Suppose further that, for some \(r>0\), the sequence \(r^ n(f^{(n)}(x)/n!)\) is bounded, for each \(x\in I\). Then f is analytic on I. Theorem 2 (E. Corominas and F. Sunyer Balaguer, 1954). Suppose \(f\in C^{\infty}(I)\). If, for each \(x\in I\), there is a nonnegative integer \(n=n(x)\) such that \(f^{(n)}(x)=0\), then f is a polynomial. In the paper under review multivariate analogues of both theorems are given as well as their improvements in rather different directions. Reviewer's remark: The authors conjecture the following: Suppose D is a connected open subset of \({\mathbb{R}}^ n\), \(f\in C^{\infty}(D)\) and that, for each \(x\in D\), there is some \(m=m(x)\in {\mathbb{N}}\) such that \(f^{\alpha}(x)=0\) for all \(\alpha \in {\mathbb{N}}^ n\) satisfying \(| \alpha | =n\). Then f is a polynomial. Note that this easily follows from Theorem 2 and from the fact that a function (on \({\mathbb{R}}^ n)\) that is a polynomial with respect to each variable separately must be a polynomial.
reviewed by: Wiesław Pleśniak
Mathematics Subject Classification ID: 26E10
zbMATH DE Number: 4120450
zbMATH Keywords: analyticity
zbMATH Keywords: polynomiality
zbMATH Keywords: infinitely differentiable functions
author: Artin B. Boghossian
author: P. D. jun. Johnson
MaRDI profile type: Publication
cites work: A theorem on analytic functions of a real variable
cites work: Q3262310
cites work: Q5827064
cites work: Q5565769
cites work: Q5569793
cites work: Singularitäten unendlich oft differenzierbarer Funktionen
full work available at URL: 0022-247x(89)90065-6
OpenAlex ID: W2006197577
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description: scientific article; zbMATH DE number 4120450