Approximation by repeated Padé approximants

It is well known that the nth diagonal Padé approximant (PA) of a Markov-Stieltjes function f converges with geometric rate to f. A special case of what we prove is that if n = 2k the same rate of convergence may be obtained by first constructing the kth diagonal PA p1q1 of f and then the kth diagonal PA of q12f.