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.
pdf version
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.
ps version,
pdf version
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.
ps version,
pdf version
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.
Advanced books
Harmonic Analysis. Real-variable methods,
orthogonality, and oscillatory integrals
by E. Stein. Princeton University Series 43, 1993.
Wavelets and Operators I, II
by Y. Meyer. Cambridge University Press, 1992.
Grades: The final grade will be determined by homework , and a project to be presented to the class.
Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.
Return to: Department of Mathematics and Statistics, University of New Mexico
Last updated: 17 January 2005