Math 545 Harmonic Analysis - Fall 2008

    The lecture notes for Math 545 Harmonic Analysis are now complete (5 January 2009).

    Topics

    • Topic 1 - Fourier coefficients: basic properties (Complete - Richard Laugesen and Aleksandra Kwiatkowska)
    • Topic 2 - Fourier series: summability in norm (Complete - Richard Laugesen)
    • Topic 3 - Fourier series: summability pointwise, Part A (Complete - Ping Xu)
    • Topic 4 - Fourier coefficients in l1 (Complete - Aleksandra Kwiatkowska)
    • Topic 5 - Fourier coefficients in l2 (Complete - Ping Xu)
    • Topic 6 - Maximal functions (Complete - Noel DeJarnette)
    • Topic 7 - Fourier series: summability pointwise, Part B (Complete - Richard Laugesen)
    • Topic 8 - Fourier series: convergence pointwise (Complete - Eunmi Kim)
    • Topic 9 - Fourier series: convergence in norm via Hilbert transform (Complete - Kostya Sluttsky)
    • Topic 10 - Hilbert transform on L2(T) (Complete - Kostya Sluttsky)
    • Topic 11 - Calderon-Zygmund decompositions (Complete - Kostya Sluttsky)
    • Topic 12 - Hilbert transform on Lp(T) (Complete - Khang Tran)
    • Topic 13 - Applications of interpolation (Complete - Richard Laugesen)
    • Topic 14 - Fourier transforms: basic properties (Complete - Richard Laugesen)
    • Topic 15 - Fourier integrals: summability in norm (Complete - Richard Laugesen)
    • Topic 16 - Fourier transform in L1(Rd), and Fourier inversion (Complete - Richard Laugesen)
    • Topic 17 - Fourier transform in L2(Rd) (Complete - Richard Laugesen)
    • Topic 18 - Fourier integrals: summability pointwise (Complete - Richard Laugesen)
    • Topic 19 - Fourier integrals: norm convergence (Complete - Richard Laugesen)
    • Topic 20 - Hilbert and Riesz transforms on L2(Rd) (Complete - Richard Laugesen)
    • Topic 21 - Hilbert and Riesz transforms on Lp(Rd) (Complete - Richard Laugesen)
    • Topic 22 - Compactly supported Fourier transforms, and the sampling theorem (Complete - Richard Laugesen)
    • Topic 23 - Periodization and Poisson summation (Complete - Richard Laugesen)
    • Topic 24 - Uncertainty principles (Complete - Richard Laugesen)

    Appendices

    • Appendix A - Minkowski's integral inequality (Complete - Richard Laugesen)
    • Appendix B - Lp norms and the distribution function (Complete - Richard Laugesen)
    • Appendix C - Interpolation (Complete - Eunmi Kim)

    These notes rearrange the material slightly from what was presented in class in Fall 2009: smoothness and decay are now treated in Topic 1, and summability has been split off into Topic 2. Homogeneous Banach spaces over T have been removed, and instead Minkowski's integral inequality is used (see Appendix A). Also, I have swapped the order of Topics 4 and 5.

    My sincere thanks go to all of you (credited above) who TeX-ed up parts of the notes!
    - Richard Laugesen