Defects and the convergence of Padé approximants (Q1591296)

Let \(f(z)\) be a function meromorphic in some domain. The Padé approximant \([L/M]\) is the unique rational function \([L/M]={P_{L,M} (z)\over Q_{L,M}(z)}\), where \(P_{L,M}(z)\) is a polynomial of degree \(\leq M\), \(Q_{L,M}(z)\) is a polynomial of degree \(\leq L\), and \(f(z)Q_{L,M}(z)= P_{L,M}(z)+ O(z^{L+M+ 1})\). It has been noted that there are functions \(f(z)\) for which the Padé approximant \([L/M]\) has a zero and a pole very close together in the domain of \(f(z)\), and this phenomenon allows for sequences of approximants that do not converge in a reasonable way on the domain of the function \(f(z)\). The author gives an extensive survey of convergence results for Padé approximants, plus one new result: if \(f(z)\) is given by a power series that is regular on the set \(\{z:|z|\leq 1,\;z\neq 1\}\), but \(f(z)\) is continuous at the point \(z=1\) (when considered as a function with domain \(\{z:|z|\leq 1\})\), then \(\lim_{L\to \infty}[L/M] =f(z)\) on \(\{z:|z|\leq 1\), \(z\neq 1\}\). From examples, the fact that there is only one point of non-regularity on the boundary of the domain seems to be essential for this result to be valid.