Math 472/572  Fourier analysis and wavelets
M ATH
472/572 
FOURIER ANALYSIS AND WAVELETS
Fall 2012

Instructor:
Cristina Pereyra
 Email:
crisp AT math . unm . edu
 Office: SMLC 320
 Phone: 2774613 (best by email)
 Schedule: TTh 9:3010:45am, in room MITCH 221
 Office Hours:Mon 34:30pm, Th 23:30pm or by appointment
 Course Assistant: TBA if any
xxx@math.unm.edu
277xxxx, SMLC xxx, Office hours: TBA
This class is crosslisted as:
 Math 472  Call # 44645  Fourier analysis and wavelets
 Math 572  Call # 44646  Fourier analysis and wavelets
(Graduate students please register in Math 572.)
Here are quick links to the homework, and to the
textbook.
This course is an introduction to Fourier Analysis and Wavelets.
It has been specifically
designed for engineers, scientists, statisticians and mathematicians interested
in the basic mathematical ideas underlying Fourier analysis, wavelets and their
applications.
This course integrates the
classical Fourier Theory with its latest offspring, the Theory of Wavelets.
Wavelets and Fourier analysis are invaluable tools for researchers in many
areas of mathematics and the applied sciences, to name a few:
signal processing, statistics, physics,
differential equations, numerical analysis, geophysics, medical
imaging, fractals, harmonic analysis, etc. It is their
multidisciplinary nature that makes these theories so appealing.
Topics will include:
 Fourier series: pointwise convergence, summability methods,
meansquare convergence.
 Discrete Fourier Transform (including Fast Fourier Transform),
and Discrete Haar Transform (including Fast Haar Transform)
 Fourier transform on the line. Timefrequency diccionary.
Heisenberg's Uncertainty Principle, Sampling theorems and
other applications. Including excursions into Lp spaces and
distributions.
 Time/frequency analysis, windowed Fourier Transform, Gabor basis,
Wavelets.
 Multiresolution analysis on the line.
Prime example: the Haar basis. Basic wavelets examples:
Shannon's and Daubechies' compactly supported wavelets.
Time permiting we will explore variations over the theme
of wavelets: Biorthogonal
wavelets, and twodimentional wavelets for image processing.
Numerical experiments are necessary to fully understand the scope of
the theory. We will let the students explore this realm according
to their interests.
The use of some Wavelet Toolbox will be encouraged.
There exists a
WAVELAB 850 package which is
Matlab
based software designed by
a team at Stanford and available for free on the Internet.
MATLAB 7.12.0 is
available in the Mathematics and Statistics Department Computer Laboratory.
Textbook: We will be use a book that
wrote with my colleague Lesley Ward from University
of South Australia and just came out. The book is called
Harmonic Analysis: From Fourier to Wavelets
, Student Mathematical Library Series, Volume 63,
American Mathematical Society 2012.
I appreciate all the feedback I can get from you in terms of typos,
erratas, and possible improvements for the second edition!
Grades: Grades will be based on homeworks,
projects and/or
takehome exams.
Prerequisites: Linear algebra and advanced calculus, or permission
from the instructor.
Recommended Texts:
The literature for Fourier Analysis and Wavelets is large.
 Good for undergraduates with strong linear algebra background:
An Introduction to Wavelets Through Linear Algebra
by
Michael Frazier Springer Verlag, Feb 1999; ISBN: 0387986391.
 More appropriate for graduate students:
Introduction to Fourier Analysis and Wavelets
by Mark A. Pinsky.
The Brooks/Cole Series in Advanced Mathematics, 2002;
ISBN 0534376606
 This book is of encyclopedic nature, excellent for graduate students
in engineering and also in statistics/math:
A Wavelet Tour of Signal Processing. The sparse way
by
S. Mallat,
Third Edition, Academic Press, 2008; ISBN 9780123743701
There are many excellent books devoted to the classical theory of
Fourier analysis (starting with Trigonometric Series by A. Zygmund
2nd edition, Cambridge University Press, Cambridge 1959,
and following with a long list).
 Appropriate for advanced undergraduate students:
Fourier Analysis: An Introduction
by E. M. Stein and R. Shakarchi,
Princeton lectures in Analysis I, Princeton University Press, 2003;
ISBN 069111384X.
 Appropriate for advanced undergraduate students, full of historical
notes and anecdotes: Fourier Analysis
by T. Korner. Cambridge University Press, 1989;
ISBN 0521389917
 A bit more advanced: Fourier series and integrals
by H. Dym and H.P. McKean.
Academic Press, 1986; ISBN: 0122264517
 A bit more advanced: An Introduction to Harmonic Analysis
by Y. Katznelson.
Dover Publications Inc. New York, NY 1976;
ISBN o486633314
In the last 1520 years there have been
published a number of books on wavelets,
as well as countless articles. Here is a limited guide:
More Mathematical
 A classic, for graduate students: Ten lectures on wavelets, by Ingrid Daubechies, 1992.
 For graduate students: A mathematical introduction to wavelets, by P. Wojtaszczyk, 1997.
 For advanced graduate students: A first course on wavelets, by E. Hernandez and G.
Weiss, 1996.
 For advanced graduate students: Wavelets and operators, by Yves Meyer, 1992.
More applied/friendlier
 Wavelets and Filter Banks, by G. Strang and
T. Nguyen, 1996.
 An introduction to wavelets, by C. K. Chui, 1992.
 A friendly guide to wavelets, by G. Keiser, 1994.
For a wider audience or emphasis on applications
 The world according to wavelets, by B. Burke Hubbard,
2nd edition, 1998.
 Wavelets: Tools for science and technology, by
S. Jaffard, Y. Meyer, R. D. Ryan, 2001.
There is a wealth of information available at
wavelet.org
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: August 20, 2012