- Wavelets, fractals, iterated function systems and operator representations: STINT advanced intensive mini-course at Centre for Mathematical Sciences Lund University

### Graduate course in math; Spring 2003:

I taught 22m303 in the '03 Spring semester (S 2003), and hope to do a repeat for the engineers in the Fall of 2004. The course will cover wavelets, and topics in mathematics with direct connections to wavelet analysis, such as spectral theory for operators in Hilbert space. You will be able to follow the material if you had 22m210, or equivalent. Selected tools from harmonic analysis will also get introduced and used in the development of the subject. [You aren't expected to know this. We will introduce the material as it is needed!] Over the past 15 years or so, wavelets have become a central tool for problems in math, as well as in its applications, e.g., in approximation theory, in subdivision algorithms, and more recently in jpeg-2000.

The advances in the subject have further relied on basic ideas from signal processing [frequency-subband filters]; and from image compression and reconstruction. Signals or images are represented by functions in one or more variables, and multiscale wavelet methods facilitate analysis algorithms that bring them into digital form (A to D), and the synthesis that reproduces images from the stored data (D to A).

Textbook:

*Wavelets through a Looking Glass: The World of the Spectrum*, O. Bratteli and P. Jorgensen, Birkhäuser, Boston, 2002. Selected chapters from the following two books will also be used:*Ten Lectures on Wavelets*, I. Daubechies, SIAM, 1992; and*Wavelets: Tools for Science & Technology*, S. Jaffard, Y. Meyer, R. D. Ryan, SIAM, 2001.Martin Vetterli's lovely web Wavelet Introduction, with visual representations of multiresolutions (Brice Lecture, Rice Univ., Sept. 19 2002: "Signal Representations: from Fourier to Wavelets and Beyond")

### 22M:034 Differential Equations (Engineering Mathematics IV):

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Work displayed on this page was supported in part by the U.S. National Science Foundation under grants DMS-9987777, DMS-0139473(FRG), and DMS-0457581.

This page was last modified on August 25, 2006 by Brian Treadway.