Dynamical sampling

TL;DR

This paper investigates the conditions under which functions can be reconstructed from coarse samples of a function and its future states under a bounded operator, providing complete solutions in finite dimensions and significant results in infinite dimensions.

Contribution

It offers a complete characterization of the dynamical sampling problem in finite-dimensional spaces and extends the analysis to infinite-dimensional spaces using advanced theorems.

Findings

Y cannot be a Riesz basis when Omega is finite.

Y is not a frame in most cases when Omega is finite.

Special cases where Y forms a frame are linked to Carleson's Theorem.

Abstract

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).