Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group (Springer Monographs in Mathematics)

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. Front Matter....Pages i-xi Front Matter....Pages 1-3 General Considerations....Pages 5-33 Analogues of the Beltrami–Klein Model for Rank One Symmetric Spaces of Noncompact Type....Pages 35-60 Realizations of Rank One Symmetric Spaces of Compact Type....Pages 61-83 Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces....Pages 85-134 Non-Euclidean Analogues of Plane Waves....Pages 135-152 Back Matter....Pages 153-156 Front Matter....Pages 157-159 Preliminaries....Pages 161-176 Some Special Functions....Pages 177-200 Exponential Expansions....Pages 201-229 Multidimensional Euclidean Case....Pages 231-267 The Case of Symmetric Spaces X = G / K of Noncompact Type....Pages 269-334 The Case of Compact Symmetric Spaces....Pages 335-370 The Case of Phase Space....Pages 371-394 Back Matter....Pages 395-399 Front Matter....Pages 401-403 Mean Periodic Functions on Subsets of the Real Line....Pages 405-440 Mean Periodic Functions on Multidimensional Domains....Pages 441-486 Mean Periodic Functions on G / K ....Pages 487-522 Mean Periodic Functions on Compact Symmetric Spaces of Rank One....Pages 523-544 Mean Periodicity on Phase Space and the Heisenberg Group....Pages 545-557 Back Matter....Pages 559-569 Front Matter....Pages 571-574 A New Look at the Schwartz Theory....Pages 575-596 Recent Developments in the Spectral Analysis Problem for Higher Dimensions....Pages 597-614 Back Matter....Pages 615-631 Front Matter....Pages 639-646 Spherical Spectral Analysis on Subsets of Compact Symmetric Spaces....Pages 571-574 Back Matter....Pages 633-638 Back Matter....Pages 639-646 ....Pages 647-671

the 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.

very Up-to-date, 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 Provide An Introductory Part Developing Analysis On Symmetric Spaces Without 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.

Annotation The 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. Very up-to-date, 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 provide an introductory part developing analysis on symmetric spaces without 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 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.