A study of refinable function vectors via a nonlinear operator

This paper studies a certain nonlinear operator T from L² (R,C^N) to itself under which every refinable function vector is a fixed point. The iterations Tⁿf of T on any f∈L²(R,C^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^mf to a limit in different topologies