Projections and dyadic Parseval frame MRA wavelets

Abstract

A classical theorem attributed to Naimark states that, given a Parseval frame B in a Hilbert space H, one can embed H in a larger Hilbert space K so that the image of B is the projection of an orthonormal basis for K. In the present work, we revisit the notion of Parseval frame MRA wavelets from [12] and [13] 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 [12] and [13].