Dynamical sampling

By Akram Aldroubi, Carlos Cabrelli, Ursula Molter, S. Tang. 2014

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Let Y={f(i), Af(i),..., A^{li} f(i): i in Omega}, where A is a bounded operator on l^2(I). The problem under consideration is to find necessary and sufficient conditions on A, Omega, {l_i:i in Omega} in order to recover any f \in l^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 M\"untz-Sz\'asz Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that Y can never be a Riesz basis when Omega is finite. We can also show that, when Omega is finite, Y={f(i), Af(i),..., A^{li}f(i): i in Omega} 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).

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8ba98c31b102ee5ca8ac5b631a49cad708af9c94

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Dynamical sampling and frame representations with bounded operators (2018, background): “uﬃciently small, and otherwise a frame sequence. Finally, we prove that iterates of a compact operator acting on a ﬁnite collection of vectors can not generate a frame. This generalizes a result from [2]. In the rest of this introduction we will collect the necessary background from frame theory and”
Frame properties of operator orbits (2018, background): “The frame characterization in the next theorem was already proved in [3,4] (see also [5] for a more general version).”
Operator Representations of Frames: Boundedness, Duality, and Stability (2017, background): “The idea of representing frames on the form (1.1) is also closely related with dynamical sampling, see, e.g., [2].”
Iterative actions of normal operators (2016, background): “6 with the result in [3], where A was assumed to be a diagonal operator on l(2)(N), we get the following characterization for a general normal operator A ∈ B(H), when |G| = 1.”