Harmonic analysis of mean periodic functions on symmetric spaces and the Heisenberg group

Prof. Valery V. Volchkov, Prof. Vitaly V. Volchkov (auth.)1848825323, 9781848825321, 9781848825338

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.

Table of contents :
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