Piecewise-smooth refinable functions

Abstract

Abstract. Univariate piecewise-smooth refinable functions (i.e., compactly supported solutions of the equation ϕ( x 2 ) = N k=0 c k ϕ(x−k)) are classified completely. Characterization of the structure of refinable splines leads to a simple convergence criterion for the subdivision schemes corresponding to such splines, and to explicit computation of the rate of convergence. This makes it possible to prove a factorization theorem about decomposition of any smooth refinable function (not necessarily stable or corresponding to a convergent subdivision scheme) into a convolution of a continuous refinable function and a refinable spline of the corresponding order. These results are applied to a problem of combinatorial number theory (the asymptotics of Euler's partition function). The results of the paper generalize several previously known statements about refinement equations and help to solve two open problems. §1. Introduction and the statement of the problem Refinement equations of the type(univariate two-scale difference equations with compactly supported mask) have found many applications in wavelets and subdivision algorithms in approximation theory, as well as in the design of curves and surfaces, in probability theory, combinatorial number theory, mathematical physics, and so on (see