Dynamical Sampling

Abstract

Let f ∈ l^2(I) be a signal at time t = 0 of an evolution process controlled by a bounded linear operator A that produces the signals A f , A^2 f , · · · at times t = 1, 2, · · · . Let Y = { f (i), Af (i), · · · , A^(l_i )f (i) : i ∈ Ω ⊂ I} be the spatio-temporal samples taken at various time levels. The problem under consideration is to find necessary and sufficient conditions on A, Ω, l_i in order to recover any f ∈ l^2(I) from the measurements Y . This is the so called Dynamical Sampling Problem in which we seek to recover a signal f by combining coarse samples of f and its futures states A^lf . Various versions of dynamical sampling problems exhibit features that are similar to many fundamental problems: deconvolution, filter banks, super-resolution, compressed sensing etc. In this dissertation, we will study these problems.