Characterization of the separativity of ultradifferentiable classes
type: Item
instance of: scholarly article
title: Characterization of the separativity of ultradifferentiable classes
zbMATH Open document ID: 0603.46043
author: Waichiro Matsumoto
published in: Journal of Mathematics of Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: Let \(\{M_ n\}^{\infty}_{n=0}\) be a logarithmicly convex and nondecreasing sequence of positive numbers. We assume that \(\lim_{n\to \infty}(M_ n)^{1/n}=\infty\). We set \({\mathcal B}_ R\{M_ n\}=\{f(x)\in C^{\infty}({\mathbb{R}}^{\ell});\| f\|_{\{M_ n\},R}\equiv \sup | f^{(\alpha)}(x)| <\infty\), where x runs over \({\mathbb{R}}^{\ell}\) and \(\alpha\) does over \({\mathbb{Z}}_+^{\ell}\}\), and \({\mathcal B}\{M_ n\}= \lim_{R\to \infty}{\mathcal B}_ R\{M_ n\}\). We call \({\mathcal B}\{M_ n\}\) the ultradifferentiable class with weight \(\{M_ n\}.\) When we consider problems on \({\mathcal B}\{M_ n\}\), the following condition sometimes plays an essential role: (S.1) \(\exists R\geq 1\), \(\forall n\), \(m\gg 1\), \(M_{n+m}\leq R^{n+m}M_ nM_ m.\) We call it the separativity condition. In this paper, we gave three equivalent conditions to (S.1). However, in order to consider a theory of pseudo-differential operators acting on \({\mathcal B}\{M_ n\}\), more various equivalent representations are required. We set \(T(r)=\sup_{n\in {\mathbb{Z}}_+}r^ n/M_ n\) \((r>0)\) and call it the associated function of \(\{M_ n\}.\) Theorem 1. The following conditions are equivalent to (S.1). (S.2) \(\exists R\geq 1\), \(\forall n\gg 1\), \((M_{2n})^{1/2n}\leq R(M_ n)^{1/n}.\) (S.3) \(\exists R\geq 1\), \(\forall n\gg 1\), \(M_{2n+1}/M_{2n}\leq RM_{n+1}/M_ n.\) (S.4) \(\exists R\geq 1\), \(\forall n\gg 1\), \(M_{n+1}/M_ n\leq R(M_ n)^{1/n}.\) (S.5) \(\exists R\geq 1\), \(\forall r\gg 1\), \(T(r)\geq T(r/R)^ 2.\) In Appendix B in [\textit{W. Matsumoto}, Theory of pseudo-differential operators of ultradifferentiable class, to appear in J. Math. Kyoto Univ.], we have proved the part which did not appear in this paper. Further, in this paper, we obtained some necessary conditions for (S.1). Theorem 2. (S.1) implies the following equivalent conditions. The converse is not always true. (S.6) \(\exists \nu >0\), \(\forall n\gg 1\), \(M_ n\leq n!^{\nu}.\) (S.7) \(\exists \nu >0\), \(\fora…
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 47Gxx
zbMATH DE Number: 3973605
zbMATH Keywords: ultradifferentiable class
zbMATH Keywords: separativity condition
zbMATH Keywords: pseudo-differential operators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250521224
OpenAlex ID: W1562440625
DOI: 10.1215/KJM/1250521224
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description: scientific article; zbMATH DE number 3973605