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Integration and Harmonic Analysis on Compact Groups

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Part of London Mathematical Society Lecture Note Series

  • Date Published: March 2011
  • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • format: Adobe eBook Reader
  • isbn: 9780511891786

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  • 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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    Product details

    • Date Published: March 2011
    • format: Adobe eBook Reader
    • isbn: 9780511891786
    • availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
  • 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.

  • Author

    R. E. Edwards

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