




Recent work: Harmonic Analysis for the Natural Sciences
Since 2010 I am developing a research program at the frontier of pure and applied mathematics with the goal of understanding the interactions between three mathematical problems arising in Natural Science (with a special emphasis in Acoustics): (1) Localization (2) Large scale (3) Spectral estimation. 
(1) Call it “The Localization Problem”. In every science where the observation process plays an important role, there is often the need of restricting the measurements to a subregion of the ambient space. The reasons can be diverse. Either the values outside that region are not accessible or they are simply irrelevant to the problem. The proper mathematical instruments to deal with this problem are called Toeplitz (or localization) operator. I am interested in continuous, discrete, Euclidean (specially in the Gabor setup), Hyperbolic (Wavelet setup) and Spherical localization. The classical approach is concerned with the spectral problem associated with Localization operators. Related problems arise in the study of geometry of sampling and interpolating sequences and the theory of frames. Included in this topic are my collaborations with P. Balazs, A. S. Bandeira, M. de Gosson, M. Dörfler, N. Faustino, H. G. Feichtinger, K. Gröchenig, Z. Mouayn, J. M. Pereira and J. L. Romero.
(2) Call it “The Large Scale Problem”. Another fundamental need of Natural Science is to understand the behavior of random phenomena in high dimensions. A big challenge is to understand the macroscopic laws of largescale mathematical and physical systems from their interaction at the microscopic level. Often such random systems display chaotic patterns in low dimensions, but if the dimension of the system is increased beyond a certain point, the systems start organizing themselves to yield familiar patterns, often approaching those arising from simple random distribution laws. The central limit theorem and the asymptotic distribution of eigenvalues of certain random matrices are the most well known examples. My prefered mathematical instrument to deal with this problem is called Determinantal Point Processes. I am interested in the large scale behaviour of processes of this type which are constructed from the eigenfunctions of localization operators, with a emphasis on universality of limit distributions, the corresponding rates of convergence and hyperuniformity in TorquatoStillinger sense. So far, this direction has been explored in collaboration with K. Gröchenig, J. M. Pereira, J.L. Romero and S. Torquato.
(3) It is called “The Spectral Estimation Problem”. Scientists often need to approximate the random laws of nature from a few deterministic observations. A fundamental problem is the one of understanding natural phenomena modeled by random models where periodic characteristics can be identified. In the collaboration with J.L. Romero we have developed a new methodology which brings together the work previously done in the above problems (1) and (2). This already lead to (as far as we know the first one!) a rigorous proof of the fact, heuristically accepted since Thomson´s seminal paper (Harmonic Analysis and Spectral Estimation, Proc.IEEE, 70, 10551095, (1982)), that the spectral window of Thomson´s multitaper approaches a perfect bandpass filter. We have obtained L1 estimates for the error in this approximation. Our estimates can be used to bound the bias of the estimator and obtain MSE bounds for several algorithms. We have a technical report outlining the proof [L. D. Abreu and J. L. Romero, On the spectral window of Thomson´s estimator, arxiv.org/abs/1503.02991 (2015).]. The results have also applications in offgrid compressed sensing using Discrete Prolate Spheroidal Sequences.

Research Centers I interact mostly: ▪ ARI (Vienna)

