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, (Postscript File) (PDF File)
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