Spectral Radius of Refinement and Subdivision Operators with Power Diagonal Dilations

Two types of estimate for the spectral radius of the multivariate refinement operator with power diagonal dilations are presented. One type contains multiplicator norm of number matrices generated by the symbol of the corresponding operator and by specific subsets of repeating fractions. These subsets are used together with the little Fermat theorem to establish estimates that comprise integrals over tori of various dimensions. Moreover, we note certain classes of symbols when the exact value of the spectral radius of refinement operator can be found. For the spectral radius of subdivision operators point value estimates are established. Submitted: April 25, 2007. Accepted: November 5, 2007.