Nonuniform cascade algorithms

For a sequence of bounded linear operators Hk, k=0,1,…, on a Banach space S, the algorithm φk,n=Hkφk+1,n−1 generates a family of sequences (φk,n)n=0∞, k=0,1,…, from an initial family of vectors φk,0∈S, k=0,1,…. We study the convergence of φk,n as n→∞, and give an application on the convergence of cascade algorithms for nonuniform splines when S is the space of all sequences φ≔(φi)i∈Z with norm ∥φ∥≔supi∈Z∥φi∥<∞, and φi, i∈Z, belong to the Banach space X=L2(R).