On Convergence and Degeneracy in Rational Padi and Chebyshev Approximation *

We study two questions associated with rational approximation of a function f(z) near the origin z-0: continuity of the Pad approximation operator, and convergence of Chebyshev to Pad ap-proximants as the domain of approximation shrinks to a point. Both become delicate in the case of degenerate approximations, i.e. approximations whose numerator and denominator are deficient in degree. In this situation various distinct definitions of convergence of sequences of rational functions make sense, and we give a unified treatment that explains their interrelationships. Our results show that the answers to the above questions are generally affirmative only in the nondegenerate case.