Projections and dyadic Parseval frame MRA wavelets

A classical theorem attributed to Naimark states that, given a Parseval frame BB in a Hilbert space HH, one can embed HH in a larger Hilbert space KK so that the image of BB is the projection of an orthonormal basis for KK. 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].