Properties of refinable functions and subdivision schemes

This thesis deals with three problems: convergence rates of subdivision schemes, spectral properties of a transition operator associated with a multivariate refinement equation, and the computation of the smoothness order of refinable functions.; Subdivision schemes consist of a class of numerically stable, easily implemented algorithms for the generation of parameterized curves and surfaces. There are many papers concerned with the convergence of subdivision schemes. Here we investigate the convergence rates of subdivision schemes and propose a method to construct modified subdivision schemes which will accelerate the convergence rates. We study the univariate case in Chapter 2 and multivariate case in Chapter 3.; In Chapter 4 of this thesis, we investigate the spectral properties of the transition operator T_a and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the refinement equation $f=Sa\in ZsfM\dot{}-a.$ In particular, we provide a detailed analysis of the spectrum of the transition operator associated with a box spline. The results are then applied to interpolatory subdivision schemes induced by box splines.; The last problem we study in this thesis is the computation of the smoothness order of refinable functions. By using the theory established in Chapter 4, we propose a method to calculate the smoothness order of refinable functions by finding the spectrum of T_b restricted to different invariant subspaces. Special attention is given to computing the smoothness order of symmetric refinable functions. Numerical computations are presented to support our theoretical analysis.