A classical theorem attributed to Naimark states that, given a Parseval frame$\mathcal{B}$ in a Hilbert space $\mathcal{H}$, one can embed $\mathcal{H}$ ina larger Hilbert space $\mathcal{K}$ so that the image of $\mathcal{B}$ is theprojection of an orthonormal basis for $\mathcal{K}$. In the present work, werevisit the notion of Parseval frame MRA wavelets from two papers ofPaluszy\'nski, \v{S}iki\'c, Weiss, and Xiao (PSWX) and produce an analog ofNaimark'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 adifferent point of view on Parseval frame MRA wavelets than that of PSWX.
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