Math 8600 :

Fall Semester 2002, 13:25-14:15 MWF, Vincent Hall 6

Instructor: Willard Miller

Office: Vincent Hall 513

Office Hours: 14:30-15:20 MW, 12:15-13:05 F

Phone: 612-624-7379

miller@ima.umn.edu, miller@math.umn.edu

**The course will be devoted to topics in applied harmonic
analysis, much of it motivated by the analysis of signals: Wavelets, the
Ambiguity Functions of Radar and Sonar, and Fractals. **Some background
in Fourier analysis will be assumed. I will begin with a brief review of
Hilbert space theory and will develop some essential results from Lebesgue
integration theory, from the point of view of the completion of an inner
product space to a Hilbert space. Group representation theory lies at the
core of the applied topics, particularly in its relationship to multi-scale
analysis and self-similarity, and I will develop group theory as
needed. Most of the course will lie on the interface between theory and
applications and we will use Matlab frequently. There will be some overlap
with the subject matter of Math 5467 (Introduction to the Mathematics of
Wavelets) but the material will be treated at a higher mathematical level
and most of the topics will be new. There will be no text. I will
make the lecture notes, and other background materials for optional
usage, available in advance on my web page.

This is an interdisciplinary course, with a strong math
core, meant for graduate students in mathematics, science and engineering.

__Lecture
Notes and Supplementary Notes for the Mathematics of Wavelets__
(Postscript File) (PDF
File)

__Lecture Notes and Background Materials on
Lebesgue Theory from a Hilbert and Banach Space Perspective, Including
an Application to Fractal Image Compression__**(Postscript
File) (PDF
File)**

** Topics in Harmonic Analysis with Applications to
Radar and Sonar,**
(

**Problem
Set #1 (Postscript File) (PDF
FILE)**

**Solutions
to Problem Set #1 (Postscript File) (PDF
FILE)**

**Problem
Set #2 (Postscript File) (PDF
FILE)**

**Solutions
to non-computational problems in Problem Set #2 (Postscript File)
(PDF
FILE)**

**Useful Links:**

**The Wavelet IDR Center
Access to the IDR Framenet Portal for analyzing data via wavelet and framelet
methods.**

**Some well known and basic fractal images: Sierpinski's
triangle, Spleenwort
fern, Von
Koch curve, Iterated
Function Systems**

**Comparison
of
continuous and discrete wavelet analysis, FFT, windowed Fourier transforms
and Wigner transforms on a single signal**