Characterization of smoothness of multivariate refinable functions in Sobolev spaces

Wavelets are generated from refinable functions by using multiresolution an alysis. In this paper we investigate the smoothness properties of multivari ate refinable functions in Sobolev spaces. We characterize the optimal smoo thness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite d imensional invariant subspace. Several examples are provided to illustrate the general theory.