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

Smooth Parseval frames for $L^2(\mathbb{R})$ and generalizations to $L^2(\mathbb{R}^d)$