The Sobolev Regularity of Refinable Functions

Refinable functions underlie the theory and constructions of wavelet systems on the one hand and the theory and convergence analysis of uniform subdivision algorithms on the other. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L₂-regularity of the refinable function φ in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function φ, the relevant invariant space is proved to be finite dimensional and is completely characterized in terms of the dependence relations among the shifts of φ together with the polynomials that these shifts reproduce. The previously known formula for this compact support case requires the further assumptions that the mask is finitely supported and that the shifts of φ are stable. Adopting a stability assumption (but without assuming the finiteness of the mask), we derive that known formula from our general one. Moreover, we show that in the absence of stability, the lower bound provided by that previously known formula may be abysmal. Our characterization is further extended to the FSI (i.e., vector) case, to the unisotropic dilation matrix case, and to even snore general setups. We also establish corresponding results for refinable distributions.