AP-frames and stationary random processes

Abstract

It is known that, in general, an AP-frame is an L2(R)-frame and conversely. Here, in part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for a Gabor system {g(t−k)eil(t−k),l∈L=ω0Z,k∈K=t0Z} to be an L2(R)-Frame in terms of Gaussian stationary random processes. In addition, if X=(X(t))t∈R is a wide sense stationary random process, we study density conditions for the associated stationary sequences {〈X,gk,l〉,l∈L,k∈K}.