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.

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Part 1; Symmetric Spaces. Harmonic Analysis on Spheres.- 1. General Considerations.- 2. Analogues of the Beltrami-Klein Model for Rank One Symmetric Spaces of Non-Compact Type.- 3. Realizations of Rank One Symmetric Spaces of Compact Type.- 4. Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces.- 5. Non-Euclidean Analogues of Plane Waves.- Comments, Further Results and Open Problems.- Part 2; Transformations with Generalized Transmutation Property Associated with Eigenfunctions Expansions.- 6. Preliminaries.- 7. Some Special Functions.- 8. Exponential Expansions.- 9. Multidimensional Euclidean Case.- 10. The Case of Symmetric SpacesX = G/Kof Noncompact Type.- 11. The Case of Compact Symmetric Spaces.- 12. The Case ofPhase Space.-Comments, Further Results and Open Problems.- Part 3; Mean periodicity.- 13. Mean Periodic Functions on Subsets of the Real Line.- 14. Mean Periodic Functions on Multidimensional Domains.- 15. Mean Periodic Functions on G/K.- 16. Mean Periodic Functions on Compact Symmetric Spaces of Rank One.- 17. Mean Periodicity on Phase Space and the Heisenberg Group.-Comments, Further Results and Open Problems.- Part 4. Local Aspects of Spectral Analysis and the Exponential Representation Problem.- 18. A New Look at the Schwartz Theory.- 19. Recent Developments in the Spectral Analysis Problem for Higher Dimensions.- 20. Spectral Analysis on Domains of Noncompact Symmetric Spaces of an Arbitrary Rank.- 21. Spherical Spectral Analysis on Subsets of Compact Symmetric Spaces.- Comments, Further Results and Open Problems.- Bibliography.