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

  • Book
  • © 2009

Overview

Authors:
  1. Valery V. Volchkov
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  2. Vitaly V. Volchkov
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  • The approach employed in this book is the best suited for dealing with the subject in a systematic fashion.
  • Most of the results are the best possible, giving answers to all questions that naturally arise in the topic and presenting the complete picture of corresponding phenomenon
  • Some significant results are published here for the first time
  • The proofs only involve concepts and facts which are indispensable to the essence of the subject
  • There is no other book available that features the same treatment of symmetric spaces

Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xi
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  2. Symmetric Spaces. Harmonic Analysis on Spheres

    1. Front Matter

      Pages 1-3
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    2. General Considerations

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 5-33

    3. Analogues of the Beltrami–Klein Model for Rank One Symmetric Spaces of Noncompact Type

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 35-60

    4. Realizations of Rank One Symmetric Spaces of Compact Type

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 61-83

    5. Realizations of the Irreducible Components of the Quasi-Regular Representation of Groups Transitive on Spheres. Invariant Subspaces

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 85-134

    6. Non-Euclidean Analogues of Plane Waves

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 135-152

  3. Back Matter

    Pages 153-156
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  4. Transformations with Generalized Transmutation Property Associated with Eigenfunctions Expansions

    1. Front Matter

      Pages 157-159
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    2. Preliminaries

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 161-176

    3. Some Special Functions

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 177-200

    4. Exponential Expansions

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 201-229

    5. Multidimensional Euclidean Case

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 231-267

    6. The Case of Symmetric Spaces X=G/K of Noncompact Type

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 269-334

    7. The Case of Compact Symmetric Spaces

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 335-370

    8. The Case of Phase Space

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 371-394

  5. Back Matter

    Pages 395-399
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  6. Mean Periodicity

    1. Front Matter

      Pages 401-403
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    2. Mean Periodic Functions on Subsets of the Real Line

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 405-440

    3. Mean Periodic Functions on Multidimensional Domains

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 441-486
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Keywords

About this book

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.

Reviews

From the reviews:

“This book is devoted to some recent developments in the harmonic analysis of mean periodic functions on symmetric spaces and Heisenberg group … . Many topics appear here for the first time in book form. The book under review was written by two leading experts who have made extensive and deep contributions to the subject in the last fifteen years. … an in-depth, modern, clear exposition of the advanced theory of harmonic analysis on the symmetric domain of rank one and the Heisenberg group.” (Jingzhi Tie, Mathematical Reviews, Issue 2011 f)

“The book is a … comprehensive research monograph, based on the author’s work. … Each section contains an introduction, notes and remarks. The book presents a modern and ambitious theme of harmonic analysis. … will mainly attract experts.” (H. G. Feichtinger, Monatshefte für Mathematik, Vol. 163 (1), May, 2011)

“The book under review is a masterly treatise whose aim is to present the theory of mean periodic functions in symmetric spaces and on the Heisenberg group … . This book is for experts in geometric analysis. … of general interest to researchers in differential geometry, analysis and probability whose work wanders into symmetric spaces. It should certainly be in the library of every university where there is research in mathematics.” (Dave Applebaum, The Mathematical Gazette, Vol. 95 (534), November, 2011)

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