Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group

This book presents the first systematic and unified treatment of the theory of mean periodic functions on homogeneous spaces. This area has its classical roots in the beginning of the twentieth century and is now a very active research area, having close connections to harmonic analysis, complex analysis, integral geometry, and analysis on symmetric spaces.

The main purpose of this book is the study of local aspects of spectral analysis and spectral synthesis on Euclidean spaces, Riemannian symmetric spaces of an arbitrary rank and Heisenberg groups. The subject can be viewed as arising from three classical topics: John's support theorem, Schwartz's fundamental principle, and Delsarte's two-radii theorem.

Highly topical, the book contains most of the significant recent results in this area with complete and detailed proofs. In order to make this book accessible to a wide audience, the authors have included an introductory section that develops analysis on symmetric spaces without the use of Lie theory. Challenging open problems are described and explained, and promising new research directions are indicated.

Designed for both experts and beginners in the field, the book is rich in methods for a wide variety of problems in many areas of mathematics.

Résumé
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 - pects of spectral analysis and spectral synthesis on homogeneous spaces. The study oftheseproblemsturnsouttobecloselyrelatedtoavarietyofquestionsinharmonic analysis, complex analysis, partial differential equations, integral geometry, appr- imation 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 ?eld of investigation at 2 the moment. The simplest examples of symmetric spaces, the classical 2-sphere S 2 and the hyperbolic plane H , play familiar roles in many areas in mathematics. The n Heisenberg groupH is a principal model for nilpotent groups, and results obtained n forH may suggest results that hold more generally for this important class of Lie groups. The purpose of this book is to develop harmonic analysis of mean periodic functions on the above spaces.

Contenu

Symmetric Spaces. Harmonic Analysis on Spheres.- General Considerations.- Analogues of the Beltrami-Klein Model for Rank One Symmetric Spaces of Noncompact Type.- Realizations of Rank One Symmetric Spaces of Compact Type.- Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces.- Non-Euclidean Analogues of Plane Waves.- Transformations with Generalized Transmutation Property Associated with Eigenfunctions Expansions.- Preliminaries.- Some Special Functions.- Exponential Expansions.- Multidimensional Euclidean Case.- The Case of Symmetric Spaces X=G/K of Noncompact Type.- The Case of Compact Symmetric Spaces.- The Case of Phase Space.- Mean Periodicity.- Mean Periodic Functions on Subsets of the Real Line.- Mean Periodic Functions on Multidimensional Domains.- Mean Periodic Functions on G/K.- Mean Periodic Functions on Compact Symmetric Spaces of Rank One.- Mean Periodicity on Phase Space and the Heisenberg Group.- Local Aspects of Spectral Analysis and the Exponential Representation Problem.- A New Look at the Schwartz Theory.- Recent Developments in the Spectral Analysis Problem for Higher Dimensions.- ????(X) Spectral Analysis on Domains of Noncompact Symmetric Spaces of Arbitrary Rank.- Spherical Spectral Analysis on Subsets of Compact Symmetric Spaces.