: A popular wavelet reference [W] states that “in theoretical and practical
studies, the notion of (wavelet) regularity has been increasing in importance.” Not surprisingly, the study of wavelet regularity is currently a major topic of investigation.
Smoother wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct wavelets converge faster for smoother wavelets. The main
goals of this paper are to extend, refine, and unify the thermodynamic approach to the
regularity of wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly
supported wavelets, and also applies to non-analytic wavelet filters.
We present an algorithm for computing the Sobolev regularity of wavelets and prove
that it converges with super-exponential speed. As an application we construct new
examples of wavelets that are smoother than the Daubechies wavelets and have the same
support. We establish smooth dependence of the regularity for wavelet families, and we
derive a variational formula for the regularity. We also show a general relation between
the asymptotic regularity of wavelet families and maximal measures for the doubling
map. Finally, we describe how these results generalize to higher dimensional wavelets