Krylov Subspace Methods in Dynamical Sampling

Let B be an unknown linear evolution process on ℂd ≃ ℓ2 (ℤd) driving an unknown initial state x and producing the states {Bℓx, ℓ = 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), …, Bℓix(i) : i ∈ Ω ⊂ ℤd}. If B is a “low-pass” convolution operator, we show that we can recover both B and x, almost surely, as long as a sufficient number of temporal samples is used. 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 method which recovers a vector with an s-sparse Fourier transform from 2s of its consecutive components.