Let Y = {f (i), Af (i), . . . , A l i f (i) : i ∈ Ω}, where A is a bounded operator on 2 (I). The problem under consideration is to find necessary and sufficient conditions on A, Ω, {li : i ∈ Ω} in order to recover any f ∈ 2 (I) from the measurements Y . This is the so called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states A l f . We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, the Müntz-Szász Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that Y can never be a Riesz basis when Ω is finite. We can also show that, when Ω is finite, Y = {f (i), Af (i), . . . , A l i f (i) : i ∈ Ω} is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H 2 (D).

Let A be a normal operator in a Hilbert space H, and let G ⊂ H be a countable set of vectors. We investigate the relations between A, G and L that make the system of iterations {A n g : g ∈ G, 0 ≤ n < L(g)} complete, Bessel, a basis, or a frame for H. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.2010 Mathematics Subject Classification. 46N99, 42C15, 94O20.