Integration and Harmonic Analysis on Compact Groups

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Overview

These notes provide a reasonably self-contained introductory survey of certain aspects of harmonic analysis on compact groups. The first part of the book seeks to give a brief account of integration theory on compact Hausdorff spaces. The second, larger part starts from the existence and essential uniqueness of an invariant integral on every compact Hausdorff group. Topics subsequently outlined include representations, the Peter-Weyl theory, positive definite functions, summability and convergence, spans of translates, closed ideals and invariant subspaces, spectral synthesis problems, the Hausdorff-Young theorem, and lacunarity.

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Table of Contents

General Introduction; Acknowledgements; Part I. Integration and the Riesz representation theorem: 1. Preliminaries regarding measures and integrals; 2. Statement and discussion of Riesz's theorem; 3. Method of proof of RRT: preliminaries; 4. First stage of extension of I; 5. Second stage of extension of I; 6. The space of integrable functions; 7. The a- measure associated with I: proof of the RRT; 8. Lebesgue's convergence theorem; 9. Concerning the necessity of the hypotheses in the RRT; 10. Historical remarks; 11. Complex-valued functions; Part II. Harmonic analysis on compact groups; 12. Invariant integration; 13. Group representations; 14. The Fourier transform; 15. The completeness and uniqueness theorems; 16. Schur's lemma and its consequences; 17. The orthogonality relations; 18. Fourier series in L2(G); 19. Positive definite functions; 20. Summability and convergence of Fourier series; 21. Closed spans of translates; 22. Structural building bricks and spectra; 23. Closed ideals and closed invariant subspaces; 24. Spectral synthesis problems; 25. The Hausdorff-Young theorem; 26. Lacunarity.

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