Operator Algebras, Representations, and Wavelets

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ABSTRACTDMS-9987777OPERATOR ALGEBRAS, REPRESENTATIONS AND WAVELETSPALLE E.T. JORGENSENJorgensen's proposal research will involve several areas of mathematics and will be focussed on applications, spectral and tiling duality, and fractal iteration processes. The basic methods derive from operator algebras and representation theory, and the connection to the applications is threefold: wavelet theory, operators, and relations between operations in the discrete and continuos domains. Wavelet theory concerns the mathematical tools involved in digitizing continuous data with view to storage, and the synthesis process, recreating the desired picture (or time signal) from the stored data. The algorithms involved go under the name of filter banks, and their spectacular efficiency derives in part from the use of (hidden) self-similarity in the data which is analyzed. Observations or time signals are functions, and classes of functions make up spaces. Numerical correlations add structure to the spaces at hand, Hilbert spaces. There are operators in the spaces deriving from the discrete data and others from the spaces of continuous signals. The first ones are good for computations, while the second reflect the real world. The operators between the two are the focus of Jorgensen's research.Relations between operations in the discrete and continuous domains are studied as symbols, because symbols are programmable. The mathematics involved in assigning operators to the symbolic relations is called representation theory. The combination of the three areas opens up exciting new opportunities at the interface of mathematics and engineering. A main point in the proposal is the study of intertwining operators between, on one side, the "discrete world" of high-pass/low-pass filters of signal processing , and on the other side, the "continuous world" of wavelets. There are significant operator-algebraic and representation-theoretic issues on both sides of the "divide", and the intertwining operators throw light on central issues for wavelets in higher dimensions. The proposal describes how the tool from diverse areas of analysis, as well as from dynamical systems and operator algebra, merge into the proposed project on wavelet analysis. The diversity of techniques is also a charm of the subject, which continues to generate new graduate student activity. Jorgensen had several recent students complete theses in the subject.