  The Riesz transform is a natural multi-dimensional extension of the Hilbert
transform, and it has been the object of study for many years due to its nice
mathematical properties. More recently, the Riesz transform and its variants
have been used to construct complex wavelets and steerable wavelet frames in
higher dimensions. The flip side of this approach, however, is that the Riesz
transform of a wavelet often has slow decay. One can nevertheless overcome this
problem by requiring the original wavelet to have sufficient smoothness, decay,
and vanishing moments. In this paper, we derive necessary conditions in terms
of these three properties that guarantee the decay of the Riesz transform and
its variants, and as an application, we show how the decay of the popular
Simoncelli wavelets can be improved by appropriately modifying their Fourier
transforms. By applying the Riesz transform to these new wavelets, we obtain
steerable frames with rapid decay.
