Principles of Harmonic Analysis
The present book is intended as a text for a graduate course on abstract harmonic analysis and its applications. The book can be used as a follow up to Anton Deitmer's previous book, A First Course in Harmonic Analysis, or independently, if the students a
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Anton Deitmar · Siegfried Echterhoff
Principles of Harmonic Analysis
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Anton Deitmar Universität Tübingen Inst. Mathematik Auf der Morgenstelle 10 72076 Tübingen Germany [email protected]
Siegfried Echterhoff Universität Münster Mathematisches Institut Einsteinstr. 62 48149 Münster Germany [email protected]
Editorial board: Sheldon Axler, San Francisco State University, San Francisco, CA, USA Vincenzo Capasso, University of Milan, Milan, Italy Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain Angus MacIntyre, Queen Mary, University of London, London, UK Kenneth Ribet, University of California, Berkeley, CA, USA Claude Sabbah, Ecole Polytechnique, Palaiseau, France Endre Süli, Oxford University, Oxford, UK Wojbor Woyczynski, Case Western Reserve University, Cleveland, OH, USA
ISBN: 978-0-387-85468-7 DOI: 10.1007/978-0-387-85469-4
e-ISBN: 978-0-387-85469-4
Library of Congress Control Number: 2008938333 Mathematics Subject Classification (2000): 43-01, 42-01, 22Bxx c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
Preface The tread of this book is formed by two fundamental principles of Harmonic Analysis: the Plancherel Formula and the Poisson Summation Formula. We first prove both for locally compact abelian groups. For non-abelian groups we discuss the Plancherel Theorem in the general situation for Type I groups. The generalization of the Poisson Summation Formula to non-abelian groups is the Selberg Trace Formula, which we prove for arbitrary groups admitting uniform lattices. As examples for the application of the Trace Formula we treat the Heisenberg group and the group SL2 (R). In the former case the trace formula yields a decomposition of the L2 -space of the Heisenberg group modulo a lattice. In the case SL2 (R), the trace formula is used to derive results like the Weil asymptotic law for hyperbolic surfaces and to provide the analytic continuation of the Selberg zeta function. We finally include a chapter on the applications of abstract Harmonic Analysis on the theory of wavelets. The present book is a text book for a graduate course on abstract harmonic analysis and its applications. The book can be used as a follow up of the First Course in Harmonic
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