Dynamical Sampling

A. ALDROUBI; C. CABRELLI; U. MOLTER; S. TANG

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2017 vol. 42 p. 378 - 401

1063-5203

Let Y={f(i),Af(i),. .,Alif(i):i∈Ω}, where A is a bounded operator on ℓ2(I). The problem under consideration is to find necessary and sufficient conditions on A,Ω,{li:i∈Ω} in order to recover any f∈ℓ2(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 Alf. 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, although Y can be complete, using the Müntz-Szász Theorem we show it can never be a basis. We can also show that, when Ω is finite, Y 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 H2(D). Finally, using the recently proved Kadison-Singer/Feichtinger theorem we show that the set obtained by normalizing the vectors of Y can never be a frame when Ω is finite.