Dynamical sampling

ALDROUBI, AKRAM; CABRELLI, CARLOS A.; MOLTER, URSULA M.; TANG, SUI

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2015

1063-5203

Let $Y={f(i), Af(i), dots, A^{li}f(i): i in Omega}$, where $A$ is a bounded operator on $ell^2(I)$. The problem under consideration is to find necessary and sufficient conditions on $A, Omega, {l_i:iinOmega}$ in order to recover any $ f in ell^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 $A^lf$. 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 M"untz-Sz´asz Theorem we show it can never be a basis. We can also show that, when $Omega$ 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 $hsd$. 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 $Omega$ is finite.