Affine AP-frames and Stationary Random Processes

TL;DR

This paper establishes a necessary and sufficient condition for affine (wavelet) systems to form AP-frames, linking stationary Gaussian processes, ergodic theory, and decay properties of associated sequences.

Contribution

It extends recent work on Gabor systems to affine wavelet systems, providing a new characterization of AP-frames via stationary Gaussian processes and ergodic theory.

Findings

Characterization of affine AP-frames using ergodic theory

Connection between decay of stationary sequences and process smoothness

Extension of Gabor system results to affine wavelet systems

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