Math 565 Syllabus

# Math 565: Introduction to Harmonic Analysis

## Instructor: Cristina Pereyra

### M ATH 565 - INTRODUCTION TO HARMONIC ANALYSIS

SRING 2005
Instructor: Cristina Pereyra
E-mail: crisp AT math . unm . edu
Office: Humanities 459, 277-4147
Schedule: TTh 2:00-3:15pm, DSH 226
Office Hours: MW 1:30-3:00pm, or by appointment

Course content: This is a first graduate course on Harmonic Analysis. Often Harmonic analysis is equated with the study of Fourier series and integrals, however the subject has evolved so much that often the most interesting problems are those where one can not use the techniques of Fourier analysis. In this course we will study classical Fourier Analysis, we will introduce the main ideas and basic techniques in modern harmonic analysis (a.k.a. the Calderon-Zygmund theory of singular integral operators), and we will discuss the basics of wavelet theory.
We will start recalling some XIX's century mathematics with Fourier's heat equation and the decomposition of functions into sums of sines and cosines. We will discuss the main question: when and how does the Fourier series of a function converges to the function itself? This already will plunge us into different modes of convergence, and in the process introduce a number of important tools, like approximations of the identity, and different averaging techniques. With the advent of measure theory new questions can be asked, convergence now can be understood in other senses (almost everywhere, in L^p, etc), and this will take us into the XX's century. We will then discuss the Fourier Transform in R^n, to do this properly we will have to introduce the Lebesque spaces L^p(R^n) (you study these throughly in a measure theory course), and distributions, as well as basic notions of Functional Analysis. We will discuss a number of basic consequences of the theory like applications to PDE's, Poisson Summation Formula, Heisenberg Principle, Radon Transform.
In the second part of the course we will introduce the main actors and universal tools used in modern harmonic analysis. The basic operators to be studied are the maximal function, Hilbert transform, square functions and paraproducts. We will note their origin in the study of classical problems in Fourier and Complex analysis, and their role as basic models for a large class of operators that appear in many problems in mathematics. One of the main problems will be to understand boundedness of these operators in L^p spaces, as well as in the limiting spaces of BMO (bounded mean oscillation) and weak L^1. We will study the basic techniques to understand boundedness of operators: Schur's Lemma, Cotlar's Lemma, interpolation and extrapolation, stopping time arguments, and show how they apply to study our basic operators.
Harmonic analysis is the art of decomposing functions and operators into simpler building blocks that can be analyzed separately and then reassembled together. The exponential functions provide an excellent analysis tool, Fourier analysis is the study of such decompositions. The trigonometric functions form an orthonormal basis in the space of square integrable functions, L^2([0,1]).. However in the line, we need a continuum of trigonometric functions to decompose a function, thus leading us to the Fourier transform (an integral). Is it possible to find orthonormal basis in the appropriate Hilbert space (L^2(R)? Yes, there are infinitely many orthonormal basis, some of the classical ones involve Hermite functions, windowed Fourier transform, etc. In the last 25 years a whole new class of bases has proved extremely useful in applications: wavelets. The theory of wavelets is know part of the curriculum in many mathematics, and engineering departments. People from different areas (physicists, engineers, harmonic analysts) arrived to it from different angles, and the fact that they communicated helped built a strong mathematical foundation with an amazing host of applications for it (including the famous FBI fingerprint standard, JPEG 2000 standard for image compression used widely in the internet, both being wavelet based). This is not a course on wavelets, however we will spend sometime learning the basic theory and understanding the basic example of Haar functions. We will compare the Haar basis to the Fourier basis, and we will study dyadic analogues of the basic operators in harmonic analysis defined in terms of the Haar basis.

Prerequisites: Real analysis (or at least advanced calculus) and linear algebra are absolutely necessary, or permission from the instructor. Notions of measure theory, functional analysis, etc. will be introduced as necessary. It will help if you have had some exposure to those more advanced ideas, but if you have not, may this course serve as an appetizer for learning in more depth those techniques.

Lecture Notes:
Lecture notes on dyadic harmonic analysis by M. C. Pereyra. Contemporary Mathematics, Volume 289 (2001) p. 1-60.
Harmonic Analysis: from Fourier to Haar by M.C. Pereyra and L. Ward. Lecture Notes delivered at the Program for Women in Mathematics at the Institute for Advanced Study, Princeton, NJ, May 2004.
Wavelets, their friends, and what they can do for you by M. C. Pereyra and M. Mohlenkamp. Lecture Notes for the short course {\em Wavelets and PDE's} at the II Panamerican Advanced Studies Institute in Computational Science and Engineering, Universidad Nacional Aut\'onoma de Honduras, Tegucigalpa, Honduras, June 2004.

Reccomended Textbooks: Depending on your interests you will find some or all of these books useful.
Fourier Analysis by Javier Duoandicoetxea. AMS Graduate Studies in Mathematics Volume 29, 2001.
Classical and Modern Fourier Analysis by Loukas Grafakos. Prentice Hall 2003.
A first course on Wavelets by E. Hernandez and G. Weiss. CRC Press, Boca Raton, FL, 1996.
A wavelet tour of signal processing by S. Mallat. Academic Press, San Diego, CA, 1998.

Expository book:
A panorama of harmonic analysis by S. Krantz. The Carus Mathematical Monographs 27, AMS 1999.
The world according to wavelets by Barbara Burke Hubbard. A K Peters Ltd., Wellesley, MA, 1996.

Classical Books
Fourier series and integrals by H. Dym, H. P. McKean.Cambridge University Press, 1987.
An introduction to harmonic analysis by Y. Katznelson. Dover Publications Inc, 1976.
Ten lectures on wavelets by I. Daubechies. CBMS-NSF Series in Applied Mathematics, SIAM, 1992.