Dynamical Sampling

MOLTER, URSULA

C.A.B.A.

Workshop; Workshop on Infinite Dimensional Analysis; 2014

The typical sampling and reconstruction problem consists of recovering a function f from its samples f(X) = {f(xj )}xj∈X. There are many situations in which the function f is an initial distribution that is evolving in time under the action of a family of evolution operators {At}t∈[0,∞): ft(x) = (Atf)(x). In some cases obtaining the samples at a sufficient rate at time t = 0 may not be possible. In this talk we present the novel method of spatio- temporal sampling in which an initial state f of an evolution process {ft}t≥0 is to be recovered from a set of samples {ft(Xt)}t∈T at differ- ent time levels, i.e., t ∈ T = {t0 = 0, t1, . . . , tN }. Clearly for the problem to be well posed, certain assumptions on f are necessary. A standard assumption (consistent with the nature of signals) is that f belongs to a Reproducing Kernel Hilbert Space (RKHS) such as a Paley-Wiener space or some other Shift Invariant Spaces (SIS) V . The objective is to describe all spatio-temporal sampling sets (X , T ) = {Xt, t ∈ T } such that any f ∈ V can be stably recovered from the samples ft(Xt), t ∈ T . We show a complete solution in the finite dimensional case, and some interesting connections with the Kadison-Singer theorem in the infinite dimensional setting.