Smoothness of Multiple Re nable Functions and Multiple Waveletsy

We consider the smoothness of solutions of a system of reenement equations written in the form as = X 2ZZ a()(2 ?) where the vector of functions = (1 ; : : : ; r) T is in (L p (IR)) r and a is a nitely supported sequence of r r matrices called the reenement mask. We use the generalized Lipschitz space Lip (; L p (IR)), > 0, to measure smoothness of a given function. Our method is to relate the optimal smoothness, p (), to the p-norm joint spectral radius of the block matrices A " , " = 0; 1, given by A " = (a("+2?)) ;; , when restricted to a certain nite dimensional common invariant subspace V. Denoting the p-norm joint spectral radius by p (A 0 j V ; A 1 j V), we show that p () 1=p ? log 2 p (A 0 j V ; A 1 j V) with equality when the shifts of 1 ; : : : ; r are stable, and the invariant subspace is generated by certain vectors induced by diierence operators of suuciently high order. This allows an eeective use of matrix theory. Also the computational implementation of our method is simple. When p = 2, the optimal smoothness is also given in terms of the spectral radius of the transition matrix associated with the reenement mask. To illustrate the theory, we give a detailed analysis of two examples where the optimal smoothness can be given explicitly. We also apply our methods to the smoothness analysis of multiple wavelets. These examples clearly demonstrate the applicability and practical power of our approach.