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