Distribution of poles of diagonal rational approximants to functions of fast rational approximability (Q1176328)

The author studies the behaviour of the poles of best rational multipoint Padé approximants (rational interpolants on a triangular array of -- not necessary distinct -- points in a compact subset \({\mathcal K}\) of the complex plane, that minimize the remainder in sup-norm on \({\mathcal K}\)) for meromorphic functions that admit fast rational approximation (i.e. the \(n\)-th root of the constant of best diagonal rational approximation converges to zero 'quickly' enough). The main results concern the existence of a subsequence of \([n_ j/n_ j]\) 'best' and 'near best' rational approximants for which the number of poles of the approximant is \(o(n_ j)\) on compact sets \({\mathcal K}\) in \({\mathcal C}\setminus{\mathcal S}\) where the function to be approximated is single-valued and analytic on \({\mathcal C}\setminus{\mathcal S}\), \(\text{cap} S=0\). The paper is an important step forward in the study of convergence properties of sequences of rational Padé-Hermite interpolants to meromorphic functions.