Math 545 Harmonic Analysis  Fall 2008 



Course outlineHarmonic 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(nt) and cos(nt), in one dimension. Immediately one asks: does the Fourier series converge? and to the original function? Convergence in what sense: pointwise? meansquare? L^{p}? Do analogous results hold on R^{d} for the Fourier transform?We will answer these classical qualitative questions (and more!) using modern quantitative estimates, involving tools such as summability methods (convolution), maximal operators, interpolation, singular integrals, Schwartz functions and distributions, uncertainty principles and LittlewoodPaley theory. These 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, bandpass filters, sampling, maximum entropy, spectral estimation and prediction). The lectures will be interspersed with such applications, as time and student interest permit. The topics covered in Fall 2008 are listed here. PrerequisitesLebesgue 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 and Lecture NotesThe lecture notes are available online, and will be your main resource. We will supplement the notes with books on reserve at the library. The main books are:"An Introduction to Harmonic Analysis" third edition (paperback $32) by Y. Katznelson; a classic text that is well worth buying. "Fourier Analysis" by J. Duoandikoetxea "Classical and Modern Fourier Analysis" by L. Grafakos You might be interested in: Software links AssessmentApproximately 4 homework assignments. No exam.Questions?Please contact me if you would like to discuss the course or its prerequisites.  Richard Laugesen Laugesen@illinois.edu 