Krylov Subspace Methods in Dynamical Sampling

Let $B$ be an unknown linear evolution process on $\mathbb C^d\simeq l^2(\mathbb Z_d)$ driving an unknown initial state $x$ and producing the states $\{B^\ell x, \ell = 0,1,\ldots\}$ at different time levels. The problem under consideration in this paper is to find as much information as possible about $B$ and $x$ from the measurements $Y=\{x(i)$, $Bx(i)$, $\dots$, $B^{\ell_i}x(i): i \in \Omega\subset \mathbb Z^d\}$. If $B$ is a "low-pass" convolution operator, we show that we can recover both $B$ and $x$, almost surely, as long as we double the amount of temporal samples needed in \cite{ADK13} to recover the signal propagated by a known operator $B$. For a general operator $B$, we can recover parts or even all of its spectrum from $Y$. As a special case of our method, we derive the centuries old Prony's method \cite{BDVMC08, P795, PP13} which recovers a vector with an $s$-sparse Fourier transform from $2s$ of its consecutive components.