Smoothness of Multiple Refinable Functions

We consider the smoothness of solutions of a system of refinement equations written in the form φ = ∑ α∈Z a(α)φ(2 · − α), where the vector of functions φ = (φ1, . . . , φr) is in (Lp(R)) and a is a finitely supported sequence of r× r matrices called the refinement mask. We use the generalized Lipschitz space Lip∗(ν, Lp(R)), ν > 0, to measure smoothness of a given function. Our method is to relate the optimal smoothness, νp(φ), to the p-norm joint spectral radius of the block matrices Aε, ε = 0, 1, given by Aε = (a(ε + 2α − β))α,β , when restricted to a certain finite dimensional common invariant subspace V . Denoting the p-norm joint spectral radius by ρp(A0|V , A1|V ), we show that νp(φ) ≥ 1/p − log2 ρp(A0|V , A1|V ) with equality when the shifts of φ1, . . . , φr are stable and the invariant subspace is generated by certain vectors induced by difference operators of sufficiently high order. This allows an effective 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 refinement 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.