MATH 7150 - Fourier Analysis

Bob Strichartz, spring 2015.

Prerequisite: MATH 6110

Texts: Stein and Weiss, Introduction to Fourier Analysis on Euclidean Spaces and Michael Taylor, Noncommutative Harmonic Analysis

Harmonic analysis = Fourier analysis deals with two basic questions: (1) How do you break up a complicated function in terms of simpler pieces? (2) How do you use the decomposition to solve problems? It plays an important role in all areas of analysis, but also has applications to many other areas of mathematics, such as probability, Lie theory, and analytic number theory.

The first part of the course, using the book of Stein and Weiss, mainly concentrates on problem (2) when (1) is the Fourier inversion formula. It develops the infrastructure of $L^p$ estimates to study basic operators like the Hilbert transform that arise in PDE theory. The second part of the course, using the book of Taylor, mainly deals with problem (1) when the functions are defined on a Lie group or a homogeneous space of a Lie group; for example, rotation groups, Lorentz groups, Heisenberg groups.

The structure of the course will involve lectures, discussions, and student presentations.