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.

In this chapter, we review some of the recent developments and prove new results concerning frames and Bessel systems generated by iterations of the form {A n g : g ∈ G, n = 0, 1, 2, . . . }, where A is a bounded linear operator on a separable complex Hilbert space H and G is a countable set of vectors in H . The system of iterations mentioned above was motivated from the so called dynamical sampling problem. In dynamical sampling, an unknown function f and its future states A n f are coarsely sampled at each time level n, 0 ≤ n < L, where A is an evolution operator that drives the system. The goal is to recover f from these space-time samples.

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We investigate systems of the form {A t g : g ∈ G, t ∈ [0, L]} where A ∈ B(H) is a normal operator in a separable Hilbert space H, G ⊂ H is a countable set, and L is a positive real number. Although the main goal of this work is to study the frame properties of {A t g : g ∈ G, t ∈ [0, L]}, as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science.2010 Mathematics Subject Classification. 46N99, 42C15, 94O20.

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