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Editors

M. Artin S.S. Chern J.M. Frohlich E. Heinz H. Hironaka F. Hirzebruch L. Hormander S. Mac Lane C.C. Moore J.K. Moser M. Nagata W. Schmidt D.S. Scott Ya.G. Sinai J. Tits B.L. van der Waerden M. Waldschmidt S. Watanabe Managing Editors

M. Berger B. Eckmann S.R.S. Varadhan

Edwin Hewitt

Kenneth A. Ross

Abstract Hartnonic Analysis Volume II Structure and Analysis for Compact Groups Analysis on Locally Compact Abelian Groups

Springer-Verlag Berlin Heidelberg GmbH

Edwin Hewitt Department of Mathematics GN-50 University of Washington Seattle, WA 98195, USA Kenneth A. Ross Department of Mathematics University of Oregon Eugene, OR 97403, USA

Second Reprint 2002

ISBN 978-3-662-24595-8 ISBN 978-3-662-26755-4 (eBook) DOI 10.1007/978-3-662-26755-4

CIP data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1970 Originally published by Springer-Verlag Berlin Heidelberg New York in 1970 41/3140-5 4 3 2 -Printed on acid-free paper SPIN 10874972

Dem Andenken John von Neumanns gewidmel

Preface This book is a continuation of Volume I of the same title [Grundlehren der mathematischen Wissenschaften, Band 115 ]. We constantly cite definitions and results from Volume 1.1 The textbook Real and abstract analysis by E. HEWITT and K. R. STROMBERG [Berlin · Gottingen ·Heidelberg: Springer-Verlag 1965], which appeared between the publication of the two volumes of this work, contains many standard facts from analysis. We use this book as a convenient reference for such facts, and denote it in the text by RAAA. Most readers will have only occasional need actually to read in RAAA. Our goal in this volume is to present the most important parts of harmonic analysis on compact groups and on locally compact Abelian groups. We deal with general locally compact groups only where they are the natural setting for what we are considering, or where one or another group provides a useful counterexample. Readers who are interested only in compact groups may read as follows: § 27, Appendix D, §§ 28-30 [omitting subheads (30.6)-(30.60)ifdesired], (31.22)-(31.25), §§ 32, 34-38, 44. Readers who are interested only in locally compact Abelian groups may read as follows: §§ 31-33, 39-42, selected Miscellaneous Theorems and Examples in §§34-38. For all readers, § 43 is interesting but optional. Obviously we have not been able to cover all of harmonic analysis. The field, already immense, is growing rapidly at the present day. We were limited