Dynamical Sampling and Systems from Iterative Actions of Operators

By Akram Aldroubi, Armenak Petrosyan. 2016

78 citations recorded by the source corpus.

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 \(\mathscr{H}\) and G is a countable set of vectors in \(\mathscr{H}\). 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bbad73389f0d0995c6a59aa60d73fd13e8dcdb21

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Frame properties of systems arising via iterated actions of operators (2019, result): “This result justifies some of the no-go results in the papers [2, 1, 3], typically of the form that certain conditions on the involved operators (including boundedness) would imply that the iterated system can not be a frame.”
Operator Representations of Frames: Boundedness, Duality, and Stability (2017, background): “For example, there exist frames {fk}k∈Z = {T f0}k∈Z where T is a unitary operator (Example 1); but if we reindex the frame as {fk}k=0 it can not be represented on the form {Uf0}n=0 for a unitary operator U, see [4].”
Dynamical sampling and frame representations with bounded operators (2018, unknown): “Further references to dynamical sampling include [2, 3, 4, 5, 13, 1, 7, 8].”