Math 715 — Fall 2001 Fourier Analysis

This course will present various aspects of Fourier analysis. We begin with review of some basic results on Fourier series. Then we introduce the Fourier transform on R^n. While discussing classical concepts such as convolution, Fourier multipliers we outline strength of Fourier analysis and its applicability in PDEs and harmonic analysis. The rest of the course will partly depend on the audience interests. We can either move in the PDE direction and discuss pseudodifferential operators and other topics connected to PDEs, or concentrate more on harmonic analysis and present classical topics such as harmonic polynomials, spherical harmonics, Paley-Wiener theory etc.

Some references:

E. Stein: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals

E. Stein, G. Weiss: Introduction to Fourier analysis in Euclidean spaces

Y. Katznelson: An introduction to Harmonic analysis

M. Taylor: Pseudodifferential operators and nonlinear PDEs