Abtastung und Identifizierung von Operatoren und Anwendungen
Zusammenfassung der Projektergebnisse
This project advanced the development of our Sampling of Operators theory. While the classical sampling theorem states that functions that are bandlimited to an interval of length Ω can be recovered from its samples taken at frequency Ω per unit interval, our sampling theory for operators considers pseudodifferential operators that have bandimited Kohn–Nirenberg symbols. In earlier work we showed, for example, that if the bandlimitation is described by a bounded Jordan domain of measure less than one, then the operator can be recovered by its action on a distribution defined on an appropriately chosen sampling grid. Within the project SamOA, we obtained explicit reconstruction formulas for the kernel of such operator. Another highlight of this project is the extension of the sampling theory to stochastic operators. Here, the bandlimitation of the Kohn-Nirenberg symbol is replaced by a support condition on the autocorrelation of the spreading function of a stochastic operator. Our results led us to the development of two estimators for the so-called WSSUS (Wide Sense Stationary with Uncorrelated Scattering) channels that are not only applicable to underspread channels, but also overspread channels with arbitrarily large, but bounded area. Our treatise of stochastic operators includes a generalization of the necessary condition for identifiability of a deterministic bandlimited operator: an operator bandimited to a Jordan domain of measure larger than one cannot be recovered through its action on any tempered distribution. In the stochastic case, this becomes a 4d volume constraint on the support of the autocorrelation of the spreading function. We show that while a 4d volume less or equal to one is necessary, it is not sufficient for identification; a crucial difference with the deterministic case where area less than one is sufficient for identification. In addition, we showed how Schwartz class functions can be used to obtain partial knowledge of operators and we obtained results on operators with small, but unknown support. Last, but not least, we discussed the finite dimensional setting in a series of papers. This problem is intimately linked to the construction of time-frequency structured measurement matrices in compressed sensing and we were able to establish performance guarantees for Basis Pursuit to recover sufficiently sparse vectors using the so obtained measurements.
Projektbezogene Publikationen (Auswahl)
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Sampling of operators. J. Four. Anal. Appl., 19(3):612, 2013
G. E. Pfander.
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The restricted isometry property for time-frequency structured random matrices. Probability Theory and Related Fields, 156(3-4):707–737, 2013
G.E. Pfander, H. Rauhut, and J.A. Tropp
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A density criterion for operator identification. Sampl. Theory Signal Image Process., 12(1):1–19, 2014
N. Grip, G. E. Pfander, and P. Rashkov
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Identification of stochastic operators. Applied and Computational Harmonic Analysis, 36(2):256 – 279, 2014
G. E. Pfander and P. Zheltov
