A study of refinable function vectors via a nonlinear operator

Abstract

This paper studies a certain nonlinear operator T from L/sup 2/(R,C/sup N/) to itself under which every refinable function vector is a fixed point. The iterations T/sup n/f of T on any f/spl isin/L/sup 2/(R,C/sup N/) with the Riesz basis property are investigated; they turn out to be the "cascade algorithm" iterates of f with weights depending on f only. The paper also gives conditions for convergence of T/sup m/f to a limit in different topologies.