Math 545 Harmonic Analysis - Fall 2008
  • Instructor: Richard Laugesen
  • e-mail: Laugesen@uiuc.edu
  • Office: 376 Altgeld Hall Phone: 333-1329
  • Office hours: TBA, or by email appointment.
  • Class: MWF 10-11am in 347 Altgeld Hall

Course outline

Harmonic analysis began with Fourier's effort to analyze (extract information from) and synthesize (reconstruct) the solutions of the heat and wave equations, in terms of harmonics. Specifically, the computation of Fourier coefficients is analysis, while writing down the Fourier series is synthesis, and the harmonics are sin(nx) and cos(nx), in one dimension.
Immediately one asks: does the Fourier series converge? and to the original function? Convergence in what sense: pointwise? L2? Lp? Do analogous results hold on Rd for the Fourier transform? We will answer these classical qualitative questions using modern quantitative estimates, involving tools such as summability methods, Schwartz functions and distributions, convolution and maximal operators, interpolation of operators and spaces, uncertainty principles and Littlewood-Paley theory.
The extension of Fourier analysis to the endpoint case p=1 yields Hardy space (and boundary values of harmonic functions) via the Hilbert transform, which motivates singular integral operator theory in higher dimensions. The above topics constitute the theoretical core of the course.

Harmonic analysis retains deep links to partial differential equations (e.g. through oscillatory integral theory) and to signal and image processing (e.g. discrete Fourier transform, windowed Fourier transform, band-pass filters, sampling, maximum entropy, spectral estimation and prediction). The lectures will be interspersed with such applications, as time and student interest permit.

Prerequisites

Lebesgue integration and L^p spaces, and beginning functional analysis (Math 541). Knowledge of complex analytic functions will be helpful too. Please talk to me if you are not sure about your background for the course.

Textbook

None. We will rely on books on reserve at the library, and lecture notes available online. The main books will be:
"An Introduction to Harmonic Analysis" third edition (paperback $32) by Y. Katznelson; a classic text that is well worth buying.
"Lectures on Harmonic Analysis" by T. H. Wolff
"Fourier Analysis" by J. Duoandikoetxea
"Harmonic Analysis and Applications" by John J. Benedetto

Assessment

Approximately 4 homework assignments. No exam.

Questions?

Please contact me if you would like to discuss the course or its prerequisites. - Richard Laugesen Laugesen@uiuc.edu