Iterates of Markov operators

AbstractIn this paper we consider iterates of Markov operators of the form Φf(x)=∑j=0mf(jmϕj(x) where the ϑj's are linearly independent, nonnegative and sum to 1. We define the evaluation matrix of Φ to be Φ∗ = [ϑj(im)] and prove that the iterates of the operator converge in the operator norm if and only if the powers of the evaluation matrix converge. Utilizing results from the theory of Markov chains we obtain explicit expressions for the limiting operator when it exists. Finally, we apply these results to Bernstein operators and then to B-spline operators