Smoothness of rank M scaling functions

Abstract

This article investigates the smoothness of rank M scaling functions and gives some numerical results. Smoothness of a scaling function ϕ is studied by introducing an exponent s p ( ϕ), which is represented by the spectral radius of the transfer operator associated with the reduced symbol of ϕ. A direct method for estimating the spectral radius is also proposed and used to obtain quite sharp estimates for s p ( ϕ) and for the Hölder exponents. In fact, numerical experiments on s 1( ϕ) for a certain class of scaling functions give estimates for their Hölder exponents much better than those obtained by the Sobolev estimates, which are known so far, combined with the Sobolev imbedding theorem.