MTH607: Euclidean harmonic analysis

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Review of basics; Fourier Series, Fourier Transforms in ${\mathbb{R}}^n$; Plancherel and Fourier Inversion Theorems. Convolutions.
• The Schwartz Space and Tempered Distributions.
• Poisson Summation Formula and applications. Uncertainty Principles: Heisenberg, Benedicks-Amrein-Berthier, and Beurling. Paley-Wiener Theorems.
• Translation Invariant Operators on $L^p$ spaces. Interpolation Theorems (Reisz-Thorin and Marcinkeiwicz).
• Maximal functions. Hilbert Transform and convergence of Fourier Series and Integrals. Calderon-Zygmund Singular Integrals.

Additional topics (a subset of the following):

• Littlewood-Paley inequalities; Hormander-Mihlin and Marcinkeiwicz Multipliers.
• $H^1$-BMO
• Time- frequency phase plane analysis. Wavelets.

Recommended Reading

• E. Stein and R. Shakarchi: Fourier Analysis, Princeton University Press (2003).
• E. Stein and R. Shakarchi: Complex Analysis, Princeton University Press (2005).
• Javier Duoandikoetxea: Fourier Analysis, AMS (2001).