Sobolev Regularity for Rank M Wavelets

This paper explores the Sobolev regularity of rank M wavelets and reenement schemes. We nd that the regularity of orthogonal wavelets with maximal vanishing moments grows at most logarithmically with lter length when M is odd, but linearly for even M. When M = 3 and M = 4, we show that the regularity does achieve these upper bounds for asymptotic growth, complementing earlier results for M = 2. A new class of wavelet lters is introduced, by asserting zeros of the wavelet symbol at preperiodic points of the mapping : ! ! M! mod 2. While this class includes the generalized Daubechies wavelets, numerical experiments demonstrate that the class also includes wavelet families with greater smoothness for a given lter length. Finally, members of the class of wavelets that have maximal Sobolev regularity for a given lter length are found as the solution to an optimization problem.