2014/09/29 by Aldroubi, A., Cabrelli, C., Molter, U. +1 · 8 citations
#42C15 #46N99 #94O20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1409.8333
Let Y=f(i), Af(i),..., Ali f(i): i in Omega, where A is a bounded operator on l2(I). The problem under consideration is to find necessary and sufficient conditions on A, Omega, li:i in Omega 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 Al 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 Müntz-Szász 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),..., Alif(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 H2(D).