Wavelets and Applications

Dear Colleagues,

Recently, there has been an enormous amount of interest in the theory and applications of wavelets. The applications include signal processing, data compression, turning fingerprints into digital data files, subdivision algorithms for graphics, and the JPEG 2000 encoding of images. As a mathematical subject, wavelet theory involves tools from a host of neighboring fields, functional and harmonic analysis, numerical analysis, mathematics of computation, and operator theory.

Wavelets now serve as an alternative to classical Fourier methods, Fourier series and integrals. The reasons for this is that they are better localized, they are better adapted to discontinuities; they have a certain
form of self-similarity, which makes the theory suited for the analysis of fractals and non-linear dynamical systems. The self-similarity properties of the scaling functions connect them to fractals and non-linear dynamical systems.

The multiresolutions offer fast algorithms. The feature of localization for wavelets is shared by related recursive basis constructions from multi-resolutions in Hilbert spaces, for example for fractals and iterated function systems in dynamics. The multiresolutions and locality yield much better pointwise approximations than is possible for traditional Fourier bases.

In signal or image-processing one is interested in subdividing analogue-signals into frequency bands. This idea goes back to Norbert Wiener, but it is of relevance in modern-day wireless signal and image processing. This suggests a representation theoretic framework. The idea is thus to build a representation theory which creates Hilbert spaces H and specific families of closed subspaces in H in such a way that "non-overlapping frequency bands" in our model correspond to orthogonal subspaces in H; or equivalently to systems of orthogonal projections. Since the different frequency bands must exhaust the signals for the entire system, one looks for orthogonal projections which add to the identity operator in H. Since time/frequency analysis is non-commutative, one is further faced with a selection of special families of commuting orthogonal projections. When an iteration scheme is applied to the initial generators, one generates new bases and frames by repeated subdivision sequences; wavelet families as recursive scheme.

Professor Palle Jorgensen
Guest Editor

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