Projections and Dyadic Parseval Frame MRA Wavelets

TL;DR

This paper extends Naimark's theorem to Parseval frame MRA wavelets, providing a new perspective on their structure and embedding properties at the level of scaling functions.

Contribution

It introduces an analog of Naimark's theorem for Parseval frame MRA wavelets, offering a self-contained and alternative viewpoint to prior work.

Findings

Established an Naimark-type theorem for Parseval frame MRA wavelets.

Provided a self-contained discussion and new perspective on the structure of these wavelets.

Connected the theory of wavelet frames with classical Hilbert space embedding results.

Abstract

A classical theorem attributed to Naimark states that, given a Parseval frame $\mathcal{B}$ in a Hilbert space $\mathcal{H}$, one can embed $\mathcal{H}$ in a larger Hilbert space $\mathcal{K}$ so that the image of $\mathcal{B}$ is the projection of an orthonormal basis for $\mathcal{K}$. In the present work, we revisit the notion of Parseval frame MRA wavelets from two papers of Paluszy\'nski, \v{S}iki\'c, Weiss, and Xiao (PSWX) and produce an analog of Naimark's theorem for these wavelets at the level of their scaling functions. We aim to make this discussion as self-contained as possible and provide a different point of view on Parseval frame MRA wavelets than that of PSWX.