Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators

Abstract

The main problem in sampling theory is to reconstruct a function from its values (samples) on some discrete subset W of its domain. However, taking samples on an appropriate sampling set W is not always practical or even possible - for example, associated measuring devices may be too expensive or scarce. In the dynamical sampling problem, it is assumed that the sparseness of the sampling locations can be compensated by involving dynamics. For example, when f is the initial state of a physical process (say, change of temperature or air pollution), we can sample its values at the same sampling locations as time progresses, and try to recover f from the combination of these spatio-temporal samples. In our recent work, we have taken a more operator-theoretic approach to the dynamical sampling problem. We assume that the unknown function f is a vector in some Hilbert space H and A is a bounded linear operator on H. The samples are given in the form (Anf,g) for 0 £ n<L(g) and g Î G, where G is a countable (finite or infinite) set of vectors in H, and the function L:G® {1,2,...,¥} represents the "sampling level." Then the main problem becomes to recover the unknown vector f Î H from the above measurements. The dynamical sampling problem has potential applications in plenacoustic sampling, on-chip sensing, data center temperature sensing, neuron-imaging, and satellite remote sensing, and more generally to Wireless Sensor Networks (WSN). It also has connections and applications to other areas of mathematics including C*-algebras, spectral theory of normal operators, and frame theory.