Nuclear function spaces on the base of a hypercomplex system
type: Item
instance of: scholarly article
title: Nuclear function spaces on the base of a hypercomplex system
zbMATH Open document ID: 0533.46012
DOI: 10.1007/BF01093154
author: A. A. Kalyuzhnyi
author: Yurij M. Berezansky
published in: Ukrainian Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: Let Q be a complete separable locally compact metric space. The authors consider socalled structure measures \(\gamma\) (A,B,r) on Q, where \(r\in Q\) and A,\(B\subseteq Q\) are Borel sets. Satisfying certain properties these measures constitute so called hypercomplex systems with basis Q, which are the spaces \(L_ 1(Q)\) with a convolution defined by \(\gamma\). Let \(\Xi\) be a fixed sequence of functions \(\xi_ n\) in \(L_ 2(Q)\) and \(D_ k=(T_{\xi_ 1}...T_{\xi_ k})^{-1}\) be ''differential'''' operators, where \(T_{\xi_ j}f=\xi_ j*f.\) These operators \(D_ k\) are defined on the range of \(T_{\xi_ 1}...T_{\xi_ k}\). Now let \({\mathcal C}^{\infty}\!_ 0(Q,\Xi)\) denote the space of all infinitely ''differentiable'' functions on Q, where differentiability is to be interpreted as above. For a certain index set \(T=\{\tau =(\tau_ 1,\tau_ 2(p)\}\) denote \(D(Q,\Xi)=\cap_{\tau \in T}H_{\tau},\) where \(H_{\tau}\) is the completion of the space \({\mathcal C}^{\infty}\!_ 0(Q,\Xi)\) with respect to the inner product \[ (f,g)_{\tau}=\sum^{\tau_ 1}_{k=0}\int(D_ kf)(p)(D_ kg)(p)\tau_ 2(p)dp,\quad \tau_ 2:Q\to [1,\infty],\quad \tau_ 1=0,1,...\quad. \] It is shown that under suitable assumptions the space D(Q,\(\Xi)\) is a nuclear space and that D(Q,\(\Xi)\) is invariant under \(T_{\xi}\).
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 46A13
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 3846216
zbMATH Keywords: nuclear function spaces
zbMATH Keywords: structure measures
zbMATH Keywords: hypercomplex systems
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3846216