# On overconvergent subsequencs of closed to rows classical Pade' approximants

## TL;DR

This paper investigates the overconvergence behavior of classical Pade' approximants for power series with positive radius of convergence, extending classical results and analyzing subsequences with specific growth conditions.

## Contribution

It extends classical overconvergence results to broader classes of Pade' approximants and subsequences with specific growth constraints on m(n).

## Key findings

- Overconvergence occurs for certain subsequences of Pade' approximants.
- Extended classical results by Hadamard and Ostrowski to new contexts.
- Provided conditions under which overconvergence is observed.

## Abstract

Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade' approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) = o(n/\log n), resp. m(n) = 0(n) as n is going to infiity. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lo'pez Lagomasino and A. Ferna'ndes Infante concerning overconvergent subsequences of a fixed row of the Pade' table.