Difference spaces and invariant linear forms
type: Item
instance of: scholarly article
title: Difference spaces and invariant linear forms
zbMATH Open document ID: 0830.43003
author: Rodney Nillsen
published in: Lecture Notes in Mathematics
publication date: 1994-12-22T00:00:00Z
review text: These Lecture Notes extensively cover the outcome of the author's long and fruitful investigations on the relationship between difference spaces and invariant linear forms. Chapter 1 presents the main ingredients and fundamental results of the subject. Let \(G\) be a locally compact group, \(1 \leq p < \infty\), and \(M(G)\) the space of bounded Radon measures on \(G\). In case \(G\) is abelian, one considers the distribution space \({\mathcal F}_p (G)\) isometrically isomorphic -- via the Fourier transform -- to \(L^p (\widehat G)\). Let \(X\) be a subspace of \(L^p(G)\) or \({\mathcal F}_p (G)\) such that \(M(G)*X \subset X\). If \(S\subset M (G)\), the difference space \({\mathcal D} (X,S)\) is defined to be the subspace of \(X\) consisting of all functions or distributions \(f\) in \(X\) such that \(f = \sum^m_{j = 1} (f_j - \mu_j *f_j)\), \(\mu_j \in S\), \(f_j \in X\), \(j = 1, \ldots, m\). A linear form \(L\) on \(X\) is said to be \(S\)-invariant if \(L(\mu*f) = L(f)\) whenever \(f \in X\) and \(\mu \in S\). Chapter II concentrating more specifically on \(R^n\) sharpens results and brings in precise properties related to Fourier transforms and wavelets. Chapter III provides applications to partial differential operators, concerning in particular Sobolev-type spaces, Hilbert and Riesz transforms. Chapter IV is devoted to \(L^p(G)\). A collection of results are established for the \(\sigma\)-compact amenable group \(G\) as it is equipped with a sequence \((K_n)\) of relatively compact neighborhoods such that \(\lim_{n \to \infty} {\mu (K_n \Delta x K_n) \over \mu(K_n)} = 0\) whenever \(x \in G\), \(\mu\) denoting a left Haar measure. The various chapters are almost self-contained. The author's statement turns out to be allright: ``A reader familiar with integration theory, Fourier theory and the theory of distributions on \(R^n\), should find Chapter II and III quite accessible. A reader familiar with basic harmonic analysis on locally compact groups should fi…
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 43A15
zbMATH DE Number: 706239
zbMATH Keywords: invariant linear forms
zbMATH Keywords: differential spaces
zbMATH Keywords: \(L^ p\)-spaces
reviewed by: Jean-Paul Pier
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 706239