Operators for Time-Frequency Analysis

Research project P 14485 Operators for Time-Frequency Analysis Karlheinz GRÖCHENIG 26.6.2000 The theory of locally compact Abelian (Ica) groups is the central part of abstract harmonic analysis (AHA) which is the well-developed mathematical art of "splitting large complicated things into small, well-understood building blocks". Mathematical engineering tools work much in the same spirit, due to the need of mapping complicated practical systems onto more simple computer-implementable models, often based on Fourier analysis. Although it would be desirable to have a solid mathematical basis for such activities (say within digital signal processing) the field of "theoretical signal analysis" (which ca be expected to emerge in the years to come) has not yet been established as a well-define research field. This is in contrast to mathematical physics with its traditional intensive interchange of problems and ideas with AHA and pure mathematics in general, however the field of wavelet theory shows clearly how a more intensive interaction between the two research areas can be of enormous benefit for both sides. In view of this situation the goal of this mathematical project is to improve the foundation3 for such an exchange on the mathematical end (but motivated by the engineering applications), and thus to help bridge the gap between the rich body of AHA and certain problems of communication and control engineering. At the same time we hope to prepare the ground for subsequent exploitation of the new insights in the form of new algorithms for signal processing or simulation of continuous systems by means of discrete approximate models. Past experience within the FSP-70 ("Mathematical Methods in Image Processing and Pattern Recognition", 1994- 2000) has shown that a good theoretical basis is the best starting point for the development of new and efficient algorithms. It is therefore expected that a similar research strategy has very good chances to be successful again in the present case. The concrete key issue for this project are operators related to time-frequency analysis resp. to the Heisenberg group that have already (implicitly) appeared in electrical engineering applications such as filter banks, non- parametric system identification and Gabor analysis, or are very likely to play a role in this area. We plan a unified and rigorous description of operators (and their continuity with respect to various norms) and algebras of such operators. While experimental evidence should help at the beginning to go in the right direction and to separate purely theoretical questions from methods with a good chance for practical applicability the results obtained during the project will be a good basis for the implementation of the most promising numerical algorithms, such as perhaps the symbolic calculus within the operator algebras mentioned. The exploitation of the full potential of those algorithms to a larger variety of applications will be carried out in subsequent, more application oriented projects.