Spectral Analysis of the Transition Operator and Its Applications to Smoothness Analysis of Wavelets

The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1, . . . , φr) be an r× 1 vector of compactly supported functions in L2(IR) satisfying the refinement equation Φ = ∑ α∈Z s a(α)Φ(M · − α), where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ1, . . . , σs) with |σ1| = · · · = |σs|. For μ = (μ1, . . . , μs) ∈ IN0, define σ−μ := σ1 1 · · ·σ−μs s . The smoothness of Φ is measured by the critical exponent