Vector Subdivision Schemes and Multiple

We consider solutions of a system of reenement equations written in the form as = X 2Z a()(2 ?) where the vector of functions = (1 ; : : : ; r) T is in (L p (R)) r and a is a nitely supported sequence of rr matrices called the reenement mask. Associated with the mask a is a linear operator Q a deened on (L p (R)) r by Q a f := P 2Z a()f(2 ?). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Q n a f) n=1;2;::: in the L p-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two nite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivisionscheme are discussed. In particular, the L 2-convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector reenement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonor-mality of multiple reenable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.