Overview

The regularity of the Meyer or the Battle-Lemari wavelets is easy to assess: the Meyer wavelet has compact Fourier transform, so that it is C ^?, and the Battle-Lemari wavelets are spline functions, more precisely, piecewise polynomial of degree k, with ( k ? 1) continuous derivatives at the knots. The regularity of compactly supported orthonormal wavelets is harder to determine. Typically, they have a non-integer H lder exponent; moreover, they are more regular in some points than in others, as is already illustrated by Figure 6.3. This chapter presents a collection of tools that have been developed over the past few years to study the regularity of these wavelets. All of these techniques rely on the fact that

where only finitely many c _n are nonzero; the wavelet ?, as a finite linear combination of translates of ?(2 x), then inherits the same regularity properties. It follows that the techniques exposed in this chapter are not restricted to wavelets alone; they apply as well to the basic functions in subdivision schemes (see 6.5). Some of the tools discussed here were in fact first developed for subdivision schemes, and not for wavelets.

The different techniques fall into two groups: those that prove decay for the Fourier transform , and those that work directly with ? itself. We will illustrate each method by applying it to the family of examples _N ? constructed in 6.4. It turns out that Fourier-based methods are...