Abstract Harmonic Analysis Volume II Structure and Analysis for Comp
This book is a continuation of vol. I (Grundlehren vol. 115, also available in softcover), and contains a detailed treatment of some important parts of harmonic analysis on compact and locally compact abelian groups. From the reviews: "This work aims at g
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Kenneth A. Ross
1\bstractlfarnaorric Analysis Volume II Structure and Analysis for Compact Groups Analysis on Locally Compact Abelian Groups
Springer-Verlag Berlin Heidelberg GmbH
ISBN 978-3-540-58318-9 ISBN 978-3-642-62008-9 (eBook) DOI 10.1007/978-3-642-62008-9
Dem Andenken John von Neumanns gelllidmet
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 J.1 The textbook Real and abstract analysis by E. HEWITT and K. R. STROMBERG [Berlin· G5ttingen· 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) if desired], (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 by space, by time, by our own abilities. We have presented the parts of the subject that every harmonic analyst must know: representations of compact groups; the WEYL-PETER theorem; PLANCHEREL'S theorem: WIENER'S Tauberian theorem. Beyond this, we have been guided largely by personal inclination. As the writing progressed, one question led naturally to another. We have omitted special topics that are not needed for our main goals and that are treated in other monographs: RUDIN [lOJ; R. E. EDWARDS [7J, [10]; KATZNELSON [3]; KAHANE and SALEM [3J; MAURIN [1]. We regret not having presented any Lie theory beyond the rudimentary facts set down in § 29. Plainly a detailed description of the continuous unitary irreducible representations of the classical compact 1 An exception is the Bibliography: every work cited in Vol. II is listed at the end of Vol. II.
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Preface
groups, going beyond e.g. BOERNER [1], and of the decompositions of their tensor products, would be of immense value for noncommutative harmonic analysis. The time seems not yet ripe for such an enterprise. A larger omission is our failure to study the algebra M(G) [G a compact or locally compact Abelian group]. This algebra presents many riddles, but enough is known for a full-scale treatment to be appropriate. We co
