Sampling and Identification of Operators and Applications

• Applicant: Professor Götz Eduard Pfander, Ph.D.
• Subject Area: Mathematics
• Term: from 2009 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 111001434

Herein, we describe recent results and future research directions in the development of a sampling theory for operators. While the classical sampling theorem asserts that functions that are bandlimited to an interval can be recovered from discrete samples taken on a sufficiently fine sampling grid, sampling results for operators allow for the recovery of pseudodifferential operators with bandlimited Kohn--Nirenberg symbols based on their response to a discretely supported distribution. In the first phase of the project Sampling and Identification of Operators and Applications (SamOA), we built on previous insights and obtained explicit reconstruction formulas for operators whose bandlimitation is described by a bounded Jordan domain of Lebesgue measure less than one. Our work includes stochastic operators where the condition that the Kohn-Nirenberg symbol is bandlimited is replaced by a support condition on the covariance of the now stochastic spreading function of an operator. Our results allowed for the development of two estimators for Wide Sense Stationary with Uncorrelated Scattering channels (WSSUS, a widely used assumption on communications channels in electrical engineering). One of these estimators is applicable to channels with arbitrarily large, but bounded support of the covariance function.In addition to the above, approximation theoretic operator sampling results were obtained as part of SamOA. These show that Schwartz class functions can be used as identifiers when we seek to obtain satisfactory information on the characteristics of bandlimited operators in relevant, time and frequency limited, transmission bands.Last, but not least, we discussed a finite dimensional analogue of the above in a series of papers. The finite dimensional identification problem is intimately linked to the construction of time-frequency structured measurement matrices for compressive sensing applications. Indeed, our results include performance guarantees for Basis Pursuit (l^1-minimization) to recover sufficiently sparse vectors using time-frequency structured measurements.Predominately, the results described above are based on regularly spaced sampling nodes (and periodic weights), that is, on identifiers supported on a lattice in Euclidean space (or mollified versions thereof). In the herein proposed continuation, we focus on irregular sampling sets. In particular, we seek to establish universal sampling sets that are applicable to operators of a given bandwidth, for example, using quasicrystals as established as universal sampling sets for functions by Meyer and Matei, or alternative universal sampling sets as those constructed by Olevskii and Ulanovskii. Moreover, we plan to examine probabilistic sampling, that is, sampling sets that are generated by random variables, and establish connections to learning theory.

• DFG Programme: Research Grants