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 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.Comment: 19 page