## Final Activity Report Summary - EUCETIFA (European centre for time-frequency analysis (Foundations, Algorithms and Applications))

The project EUCETIFA at the University of Vienna was funded by the Marie Curie Excellence grant MEXT-CT 2004-517154. The research on the project was carried out at the interface between basic mathematical research in time-frequency analysis and application-oriented topics motivated by wireless communications and data processing.

The following three topics were the main focus of the project: time-varying systems and time-frequency methods for pseudodifferential operators, sampling problems and algorithms for time-frequency representations, and the analysis and application of redundant representations of functions (frame theory and Banach algebras).

The time-frequency approach to pseudodifferential operators, as developed in the project, turned out to be especially useful for time-varying communication systems, such as wireless communications. The main achievement consists of the efficient representation of time-varying systems by Gabor frames. On the one hand, this representation had several applications in mathematics (almost diagonalisation of time-varying systems, new Banach algebras of pseudodifferential operators), on the other hand, the underlying algorithms led to a significant improvement of the equalisation procedure in wireless communications. This new method was granted a European patent on a 'Multicarrier signal equalising method of intercarrier interference'.

In addition, time-frequency methods were shown to be highly relevant in modern quantum mechanics: modulation space norms serve as quantitative measures for phase-space concentration, and the time-frequency analysis of pseudodifferential operators provided new insight into quantisation procedures and the spectrum of Schroedinger operators.

In sampling theory, the major accomplishments were the thorough analysis of Gabor frames with Hermite functions; a new theory of random sampling of band-limited functions; and a dramatic improvement of the algorithms for the reconstruction of algebraic polynomials from Fourier coefficients (the so-called inverse polynomial reconstruction method). Gabor frames with Hermite functions provided redundant time-frequency representations with functions of small uncertainty, random sampling helped to prove general results about band-limited signals in higher dimensions that were out of reach of deterministic techniques.

In the work package on frames the team has investigated a new class of adaptive algorithms that work with flexible but redundant representations of data (so-called frames) and achieved an almost optimal rate of convergence and numerical complexity. The team further studied the relation between infinite-dimensional problems and the corresponding finite-dimensional models that arise in the numerical discretisation. The main results were accurate quantitative convergence estimates for the so-called finite section method.

The total scientific output of the project consists of 64 research publications in top ranked mathematical, as well as in engineering and physics journals. Further achievements include one book, several contributed book chapters, software, and three PhD theses.

The EUCETIFA team has organised two international conferences on time-frequency analysis and time-frequency methods in Strobl near Salzburg, Austria (2007 and 2009), with an attendance of 100 - 120 scientists. The team further coorganised a conference on sampling theory and applications 2009 in Marseille and held a final project workshop at the prestigious Institute of Science and Technology, Austria.

Last not least, the European Centre for Time-Frequency Analysis has had a significant impact on the local scientific environment in Vienna and helped to attract several additional projects and grants.

The following three topics were the main focus of the project: time-varying systems and time-frequency methods for pseudodifferential operators, sampling problems and algorithms for time-frequency representations, and the analysis and application of redundant representations of functions (frame theory and Banach algebras).

The time-frequency approach to pseudodifferential operators, as developed in the project, turned out to be especially useful for time-varying communication systems, such as wireless communications. The main achievement consists of the efficient representation of time-varying systems by Gabor frames. On the one hand, this representation had several applications in mathematics (almost diagonalisation of time-varying systems, new Banach algebras of pseudodifferential operators), on the other hand, the underlying algorithms led to a significant improvement of the equalisation procedure in wireless communications. This new method was granted a European patent on a 'Multicarrier signal equalising method of intercarrier interference'.

In addition, time-frequency methods were shown to be highly relevant in modern quantum mechanics: modulation space norms serve as quantitative measures for phase-space concentration, and the time-frequency analysis of pseudodifferential operators provided new insight into quantisation procedures and the spectrum of Schroedinger operators.

In sampling theory, the major accomplishments were the thorough analysis of Gabor frames with Hermite functions; a new theory of random sampling of band-limited functions; and a dramatic improvement of the algorithms for the reconstruction of algebraic polynomials from Fourier coefficients (the so-called inverse polynomial reconstruction method). Gabor frames with Hermite functions provided redundant time-frequency representations with functions of small uncertainty, random sampling helped to prove general results about band-limited signals in higher dimensions that were out of reach of deterministic techniques.

In the work package on frames the team has investigated a new class of adaptive algorithms that work with flexible but redundant representations of data (so-called frames) and achieved an almost optimal rate of convergence and numerical complexity. The team further studied the relation between infinite-dimensional problems and the corresponding finite-dimensional models that arise in the numerical discretisation. The main results were accurate quantitative convergence estimates for the so-called finite section method.

The total scientific output of the project consists of 64 research publications in top ranked mathematical, as well as in engineering and physics journals. Further achievements include one book, several contributed book chapters, software, and three PhD theses.

The EUCETIFA team has organised two international conferences on time-frequency analysis and time-frequency methods in Strobl near Salzburg, Austria (2007 and 2009), with an attendance of 100 - 120 scientists. The team further coorganised a conference on sampling theory and applications 2009 in Marseille and held a final project workshop at the prestigious Institute of Science and Technology, Austria.

Last not least, the European Centre for Time-Frequency Analysis has had a significant impact on the local scientific environment in Vienna and helped to attract several additional projects and grants.