  Wavelet set wavelets were the first examples of wavelets that may not have
associated multiresolution analyses. Furthermore, they provided examples of
complete orthonormal wavelet systems in $L^2(\mathbb{R}^d)$ which only require
a single generating wavelet. Although work had been done to smooth these
wavelets, which are by definition discontinuous on the frequency domain,
nothing had been explicitly done over $\mathbb{R}^d$, $d >1$. This paper, along
with another one cowritten by the author, finally addresses this issue.
Smoothing does not work as expected in higher dimensions. For example, Bin
Han's proof of existence of Schwartz class functions which are Parseval frame
wavelets and approximate Parseval frame wavelet set wavelets does not easily
generalize to higher dimensions. However, a construction of wavelet sets in
$\hat{\mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown
that a commonly used class of functions cannot be the result of convolutional
smoothing of a wavelet set wavelet.
