Convergence of Vector Subdivision Schemes in Sobolev Spaces

AbstractIn this paper we consider functional equations of the form Φ=∑α∈Zsa(α)Φ(M·−α), where Φ=(φ1,…,φr)T is an r×1 vector of functions on the s-dimensional Euclidean space, a(α), α∈Zs, is a finitely supported sequence of r×r complex matrices, and M is an s×s isotropic integer matrix such that limn→∞M−n=0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φj, j=1,…,r, belonging to the Sobolev space Wkp(Rs)? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Qa associated with the sequence a is defined by QaF:=∑α∈Zsa(α)F(M·−α),F∈(Wkp(Rs))r. Let Φ0 be a nontrivial r×1 vector of compactly supported functions in Wkp(Rs). The iteration scheme Φn=QaΦn−1, n=1,2,…, is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0)=m−1∑α∈Zsa(α), m=∣detM∣, and are related to polynomial reproducibility and the classical Strang–Fix conditions.The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution Φ). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang–Fix type conditions that determine the feasible set of initial vectors