Abstract Harmonic Analysis: Structure and analysis for compact groups, analysis on locally compact Abelian groupsVolume two. 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 I.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 [omittingsubheads (30.6)-(30.6o)ifdesired], (31.22)-(31.25), §§ 32, 34-38, 44. Readers who are interested only in locally compact Abelian groups may read as follows: § § 31-3 3, 39-42, selected Miscellaneous Theorems and Examples in §§ 34-38. For all readers, § 43 is interesting but optional. |
Contents
Preface VII | 1 |
More about representations of compact groups | 60 |
Miscellaneous facts about representations | 115 |
Copyright | |
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Abstract Harmonic Analysis: Structure and Analysis for Compact Groups ... Edwin Hewitt,Kenneth A. Ross No preview available - 1970 |
Common terms and phrases
approximate unit arbitrary B₁ Banach algebra belongs Borel Borel measurable bounded linear character group compact Abelian group compact subset complex numbers contains continuous cosets defined denote direct sum dual object E₁ element equal equivalent finite finite-dimensional follows Fourier transform function f function on G ƒ ƒ G₁ group G H₁ H₂ Haar measure harmonic analysis Hence Hilbert space holds implies inequality isomorphic L₁ L₂ G Lemma Let f Let G linear functional linear subspace locally compact Abelian locally compact group mapping Math matrix nonnegative nonvoid nonzero norm normal subgroup notation operator PLANCHEREL'S theorem positive integer positive-definite functions Proof prove RAAA real number representation space sequence shows Sidon set space H spectral set spectral synthesis subset of G Suppose theorem topological unitary representation write y₁ Σ Σ σΕΣ