CONSTRUCTION AND REGULARITY OF SCALING FUNCTIONS

Using spectral properties of the operators P(w)f(x) = w(x/2)f(x/2)+ w(x/2+1/2)f(x/2+1/2), we study the scaling functions phi associated to filters with infinite length. We compute the Sobolev coefficient of phi and more generally the larger coefficient s such that integral (+infinity)(-infinity)\phi(lambda)\(p)(1+\lambda\(ps))d lambda < +infinity, where 1 less than or equal to p < +infinity. We apply the results to dyadic interpolations.