Affine AP-frames and Stationary Random Processes

Abstract

It is known that, in general, an affine or Gabor AP-frame is an $L^2(\mathbb{R})$-frame and conversely. In part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for an affine (wavelet) system $\mathcal{A}=\{a^{j/2}{\psi }_{j,k}(t):=a^{-j/2}\psi (a^{-j}t-k):j\in \mathbb{Z},k\in \mathbb{K}:=b\mathbb{Z}\}$ to be an affine AP-Frame in terms of Gaussian stationary random processes expanding in this way what we have done recently for Gabor systems. Likewise, we study a connection between the decay of the associated stationary sequences $\{⟨X,{\psi }_{j,k}⟩:k\in \mathbb{K}\}$ for each $j\in \mathbb{Z}$, and a smoothness condition on a Gaussian stationary random process $X=(X(t){)}_{t\in \mathbb{R}}$.