Colloquium

Let \(f\) an initial state of a dynamical process controlled by an operator \(A\) that produces the states \(Af, A^2f,\dots\) at times \(t=1,2,\dots\). Let \(M\) be a measurements operator applied to the series \(Af, A^2f,\dots\) at times \(t=1,2,\dots\). The problem is to recover \(f\) from the measurements \(Y=\{Mf, MAf, MA^2f, \dots,MA^Lf\}\). This is the so called Dynamical Sampling Problem. A prototypical example is when \(f \in ll^2(\mathbb Z)\), \(X\) a proper subset of \(\mathbb Z\) and \(Y=\{f_X, (Af)_X, (A^2f)_X, \dots,(A^Lf)_X\}\) where \(g_X\) denotes the restriction of \(g\) to \(X\) (that is, we erase the values of \(g\) outside \(X\)). The problem is to find the necessary and sufficient conditions on \(A\), \(X\), \(L\), that are sufficient for the recovery of \(f\).

We will discuss the problem, its applications, and some of the recent results. Typical applications are time-space sampling tradeoff (e.g., sound fields acquisition using microphones), super-resolution, on-chip sensing (e.g., sensors on chip for accurate measements of temperature, voltages, etc.), intra-cortical telemetry, source localization, and satellite remote sensing.