Institute of Mathematics

Lecture courses/details

Harmonic Analysis
Prof. Dr. Klaus Widmayer


10:15 - 12:00
Y27H12 Seats: 50

Review of topics

  • Week 1: Introduction/Historical Background. Fourier series on the torus. Basic properties of Fourier coefficients (Riemann-Lebesgue lemma). Partial sums and the Dirichlet kernel. Criteria for pointwise convergence (Dini).
  • Week 2: Partial sums and continuity (duBois-Reymond). Summability methods: Fejér and Poisson kernels. Approximate identities. Density of trigonometric polynomials in L^p.
  • Week 3: Norm convergence of partial sums: L^2 setting. Relation to boundedness of partial summation. The Maximal function: introduction and basic properties.
  • Week 4: Weak L^1 and L^p boundedness of the maximal function
  • Week 5: The maximal function and a.e. convergence of radially bounded approximate identities
  • Week 6: Weak L^p spaces, maximal operators and a.e. convergence, Marcinkiewicz interpolation
  • Week 7: Fourier transform, Schwartz space
  • Week 8: Tempered distributions and their Fourier transform
  • Week 9: Hilbert transform, boundedness on L^p (1<p<\infty)
  • Week 10: Convergence of partial sums, Calderon-Zygmund decomposition
  • Week 11: Calderon-Zygmund decomposition and weak (1,1) bounds for the Hilbert transform
  • Week 12: Truncated Hilbert transform and a.e. convergence. Introduction to singular integrals and Calderon-Zygmund operators
  • Week 13: Fractional integration operators, classical singular integrals, Calderon-Zygmund operators
  • Week 14: Calderon-Zgymund operators, outlook

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