The existence and convergence of subsequences of Padé approximants

We prove the existence of an infinite number of Padé approximants, and thereby remedy a defect in Nuttall's theorem. It is proved that the sequences of Padé approximants shown by Perron, Gammel, and Wallin to be everywhere divergent contain subsequences which are everywhere convergent. It is further proved that there always exist, for entire functions, everywhere convergent 1, N] and 2, N] subsequences of Padé approximants. There must exist subsequences of m, N] Padé approximants (N → ∞) which converge almost everywhere in $\text{¦z¦ ? ? < R}$ to functions f(z) which are regular except for a finite number (n ? m) of poles in $\text{¦z¦ < R}$. We prove convergence of the N, N + j] Padé approximants in the mean on the Riemann sphere for meromorphic functions.