Affine AP-Frames and Stationary Random Processes

Abstract

It is known that, in general, an affine or Gabor AP-frame isan L 2 (R)-frame and conversely. In part as a consequence of the ErgodicTheorem, we prove a necessary and sufficient condition for an affine(wavelet) system A = {a j/2 ψ j,k (t) := a −j/2 ψ(a −j t − k) : j ∈ Z, k ∈K := bZ} to be an affine AP-Frame in terms of Gaussian stationaryrandom processes expanding in this way what we have done recently forGabor systems. Likewise, we study a connection between the decay ofthe associated stationary sequences {⟨X, ψ j,k ⟩ : k ∈ K} for each j ∈ Z,and a smoothness condition on a Gaussian stationary random processX = (X(t)) t∈R .