Unitary matrix functions, wavelet algorithms, and structural properties of wavelets (original) (raw)
Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet constructions had previously been known, since Haar in 1910, it was the advent of multiresolutions, and sub-band filtering techniques which provided the tools for our ability to now easily create efficient algorithms, ready for a rich variety of applications to practical tasks. Part of the underpinning for this development in wavelet analysis is operator theory. This will be presented in the lectures, and we will also point to a number of developments in operator theory which in turn derive from wavelet problems, but which are of independent interest in mathematics. Some of the material will build on chapters in a new wavelet book, co-authored by the speaker and Ola Bratteli, see http://www.math.uiowa.edu/˜jorgen/. Contents Abstract 1 1. Introduction 3 1.1. Index of terminology in math and in engineering 4 1.1.1. Some background on Hilbert space 1.1.2. Connections to group theory 1.1.3. Some background on matrix functions in mathematics and in engineering 1.
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