Overview

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.

Full Product Details

Author:   Valery V. Volchkov ,  Vitaly V. Volchkov
Publisher:   Springer London Ltd
Imprint:   Springer London Ltd
Edition:   2009 ed.
Dimensions:   Width: 15.50cm , Height: 3.60cm , Length: 23.50cm
Weight:   2.490kg
ISBN:  

9781848825321

ISBN 10:   1848825323
Pages:   671
Publication Date:   17 July 2009
Audience:   College/higher education ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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.

Reviews

Aus den Rezensionen: ! die wesentlichen Grundlagen der Theorie wurden von Laurent Schwartz etabliert. ! Eigenschaften und Charakterisierungen aller im Mittel periodischen Funktionen sind von Interesse. ! ist in vier Abschnitte unterteilt. ! eine sorgfaltig geschriebene, systematische und ! konzise. ! Hilfsresultate sind sicherlich nutzlich, und sie sind ein Indikator fur die mathematische Stringenz der Herangehensweise. ! Lobend hervorzuheben ist der klare Aufbau des Buchs. Jeder der vier Teile beginnt mit einer kurzen Einfuhrung und endet mit ausfuhrlichen Kommentaren, Anknupfungspunkten und offenen Fragen. (in: Jahresbericht Deutsche Mathematiker-Vereinigung DMV, 2010, Vol. 112, Issue 2)

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 fur 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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