The theory of mean periodic functions is a subject which goes back to works of Littlewood, Delsarte, John and that has undergone a vigorous development in recent years. There has been much progress in a number of problems concerning local aspects of spectral analysis and spectral synthesis on homogeneous spaces. The study of these problems turns out to be closely related to a variety of questions in harmonic analysis, complex analysis, partial differential equations, integral geometry, approximation theory, and other branches of contemporary mathematics.
The present book describes recent advances in this direction of research. Symmetric spaces and the Heisenberg group are an active field of investigation at the moment. The purpose of this book is to develop harmonic analysis of mean periodic functions on the above spaces. The book consists of four parts. Part I is devoted to symmetric spaces and related questions. After some general considerations in Chapter 1, rank one symmetric spaces play here a privileged role. The text is based on realizations of rank one spaces as domains in Euclidean space. The aim of such an approach is twofold: on the one hand, in this way it is possible to contribute towards a better visualization and a better handling of these spaces; on the other hand, in addition to their intrinsic interest, these realizations will play an important role in our study of transmutation operators on rank one compact symmetric spaces in Part II. Part II develops the transmutation operator theory. The appropriate analogues of the Abel-Radon transform with their basic properties are defined. The generalized homomorphism property is the crucial one; it relates the mean periodicity on the spaces in question to that on \({\mathbb{R}}^1\) and allows many proofs in Parts III and IV to be carried out by reduction to the one-dimensional case. Parts III and IV deal with the theory of mean periodic functions on domains of Euclidean spaces, Riemannian symmetric spaces, and the Heisenberg group. Attention is focused on Fourier-type decompositions and on the ‘hard analysis’ problems that could be attacked with them: structure of zero sets of mean periodic functions and modern versions of John’s support theorem, local analogues of the Schwartz fundamental principle, the problem of mean-periodic continuation, Hörmander-type approximation theorems on domains without the convexity assumption, explicit reconstruction formulae in the deconvolution problem, Zalcman-type two-radii problems on domains of symmetric spaces of arbitrary rank, local versions of the Brown-Schreiber-Taylor theorem on spectral analysis and their symmetric space analogues, and so on. The proofs given are ‘minimal’ in the sense that they involve only such concepts and facts which are indispensable for the essence of the subject. All the necessary information is given in the text with references to the sources.