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description of the scientific content:The project Numerical and Applied Harmonic Analysis (naha) intends to position the Numerical Harmonic Analysis Group »NuHAG of the »Faculty of Mathematics, and hence the »University of Vienna, internationally as a prominent center of computational harmonic analysis. Harmonic Analysis interacts with several branches of the mathematical sciences, in a search for theoretical foundations and computational methods which Existing international contacts and local collaborations, especially with the Technical University of Vienna, are likely to foster fruitful. The effort is based on several recently funded major projects (in particular »EUCETIFA and »MOHAWI), which aim to study fundamental mathematical problems directly related to applications. It entails the development of new theoretical concepts (based on Banach frames, localization theory, Wiener pairs, flexible Gabor transforms, quilted Gabor frames, etc., cf. below for details), and their subsequent transformation into practical algorithms for real world applications in communication theory, medical or technical signal processing. Computational methods have played a crucial role at the »NuHAG already for many years. One of the first topics pursued intensively within the »NuHAG was the so-called irregular sampling problem (resp. scattered data approximation problem): i.e. the reconstruction of band-limited signals from samples taken at irregularly located positions. First Feichtinger-Gröchenig established a sound theoretical basis for this problem ([37][38]), then several »NuHAG members helped to demonstrated the ability to develop efficient algorithms, with good performance and useful for engineers and geophysicists (Ph.D. M. Rauth and [58] [39] [41] [4] [77] [63] ). Later on substantial contributions to Gabor analysis have been obtained at the »NuHAG by developing several advanced computational methods (e.g. algorithms to calculate dual or tight Gabor atoms in cooperation with the engineers, such as [76] [79] [78] [87] ) and theoretical understanding (the role of Janssen's representation of the Gabor frame operator, twisted convolution, adjoint groups and their role for non-separable time-frequency lattices, Gabor multipliers, see [30] [36] [35] [54] [40] [45] ). The last few years have seen various break-throughs on the theoretical side (in particular related to the work of Gröchenig an his coauthors, but also Feichtinger and Kaiblinger, etc., see [32] [43] [44] ), while computational efforts have not advanced at a similar pace. Since Prof. Gröchenig just accepted a position as a permanent faculty member, it is an excellent opportunity to make a strong effort concerning more computationally intensive work, closely connected with the ongoing theoretical research. Given the longstanding cooperation with Gröchenig, the experiences within the team and the results obtained in the last few years, we can make substantial progess within short time.
The proposed naha project will serve as a concentration point, giving the computational
side of »NuHAG a strong momentum, and allowing to expand in a systematic way
existing MATLAB code. The key topics of research will be
[ad 1] The effort of this direction is to reinforce algorithmic research carried out at »NuHAG for many years. ^ Similarly to the situation at the beginning of »NuHAG in the early 90's, theoretical investigations have outpaced software implementations. For example, stability of reconstruction methods against the jitter error has been shown, which gives rise to the possibility of reconstructing a band-limited or spline-type function from sufficiently many local averages taken at sufficiently many positions (see [1] [2] [82] [84] ). Problems arising in multi-sensor networks and the 'super-resolution problem' (obtaining a sharp image or precise information about a multi-dimensional signal from a relatively large collection of possibly imprecise measurements) require good models, the ability to handle the subtleties of the structure of the data, and understanding of regularization methods which have to be used on order to overcome effects arising from noise. We consider the regular Gabor problem (recovery of signals from sampled short-time Fourier transforms over a lattice, using dual Gabor windows), but also from irregular sampling sets in phase space, or the related problem of irregular sampling of a wavelet transform in the upper half-plane as special i nstances, where the last year has seen a bulk of literature describing qualitative results. For example, it is known, that one has different forms of stability of the problem (continuous dependence of the data and the reconstructed signal from the Gabor window and the sampling set which is used, see e.g. [43] [52] [85] ). These qualitative results ask for quantitative exploration (in practice one needs to know the actual amount of error acceptable in order to have an error estimate), as one of the themes where theory and numerical realization (by itself an interesting challenge) meet and reinforce each other. On the other hand, it will be possible to derive (as it was the case with the classical irregular sampling algorithms developed by »NuHAG members, such as [58] [41] ) error estimates (for example depending on the quality of the data and the number of iterations allowed). This is the basis for the development of real-time algorithms on a sound theoretical basis, because only this can guarantee satisfactory performance in challenging regimes. Moreover, the recently developed localization theory for Banach frames ( [59] [55] ) provides an extra tool in this direction, which is already exploited in a recent preprint by Fornasier ( [19] ). The numerical study of so-called flexible Gabor transform (used for the characterization of alpha-modulation spaces), and the discretization problem (cf. [56] ) there, are important special cases of what we have in mind. [ad 2] The very successful description of pseudo-differential operators using the Kohn-Nirenberg calculus as well as the spreading function description of time-variant linear systems (linear operators from the Schwartz space to the tempered distributions, in a mathematical terminology, see [74] ) has opened a way to discuss pseudo-differential operators using time-frequency methods ( [66] [60] ). This approach has been exploited in many ways in order to obtain theoretical results, cf. several papers by K. Gröchenig and coauthors given in the bibliography. We also expect that a number of ideas developed by Feichtinger and Ferenc Weisz in the last 2 years (classical summability methods in the context of Gabor analysis, see e.g. [53] ) will find their applications in the numerical context to be investigated within naha. Encouraging first results (again mostly at the qualitative level) are found in the recent papers [70] [72] . [ad 3] Sparse data representations, sparse recovery and computational frame theory. In ubiquitous applications, e.g., coding, digital broadcasting, or information transmission, one uses redundant data. One useful mathematical framework for redundant data is frame theory. Usually the goal is to find the sparsest representation with respect to a given frame, i.e., the one with the fewest non-zero coefficients. Closely related is the sparse recovery problem, where one tries to reconstruct a signal having a sparse representation in some basis or frame from very incomplete information. Both approaches lead to underdetermined systems of equations, what until recently was considered a disadvantage of frames (as opposed to bases). Recently mathematicians around D. Donoho, A. Gilbert, E. Candes and T. Tao have developed novel methods for treating these problems [11] [12] [26] [25] [57] [81] [86] , and these methods will likely lead to new applications. In particular, large scale computations with frames seem possible now. One of the goals of this project is an investigation of sparse time-frequency representations and related sparse recovery problems, as well as adaptive algorithms for their efficient computation. |
