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 $\{ \langle{ X,\psi_{ j,k}}\rangle : k\in\mathbb{ K}\}$ for each $j\in\mathbb{ Z}$, and a smoothness condition on a Gaussian stationary random process $X=(X(t))_{ t\in\mathbb{ R}}$.
Categories: math.PR math.FA