Krylov Subspace Methods in Dynamical Sampling

Let B be an unknown linear evolution process on C ' `(Zd) driving an unknown initial state x and producing the states {Bx, ` = 0, 1, . . .} 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), . . . , Bix(i) : i ∈ Ω ⊂ Z}. 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 [4] 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 [5, 7, 12] which recovers a vector with an s-sparse Fourier transform from 2s of its consecutive components.