On the limits of (linear combinations of) iterates of linear operators

We give necessary and sufficient conditions such that iterates or certain linear combinations of iterates of linear operators of finite dimensional range, respectively, converge. In case of convergence, we give an expression for the limit as well as estimates for the rate of convergence. Our results are then applied to Schoenberg type and Sablonniere operators as well as tensor product Schoenberg type operators, Bernstein and Bernstein-Durrmeyer operators over triangles. (C) 1997 Academic Press.