Difference spaces and invariant linear forms
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 find Chapters I and IV readily accessible.'' The volume furnishes exceptionally carefully formulated explanations and developments. Historical references and supplementary comments are added to each chapter.
scientific article; zbMATH DE number 706239