A Complete Solution Characterizing Smooth Refinable Functions

Abstract

In this paper we study univariate two-scale refinement equations $\varphi(x)=\sum_{k \in \zed} c_k\varphi\break(2x-k)$, where the coefficients $c_k \in \cx$ satisfy an exponential decay assumption. We show that any refinement equation that has a smooth solution can be reduced to the well-studied case of complete sum rules: $\sum_k(-1)^kk^nc_k = 0, n=0, . . . L, where L depends on regularity of the solution. This result makes it possible to extend previously known results on refinable functions and subdivision schemes from the case of complete sum rules to the general case. As a corollary we obtain sharp necessary conditions for the existence of smooth refinable functions and the convergence of corresponding cascade algorithms. Other applications concern polynomial spaces spanned by integer translates of a refinable function and one special property of linear operators associated to refinement equations.