Diagonal Padé approximants to hyperelliptic functions
This paper is a step forward in the theory of convergence of diagonal Padé approximants connected with a.o. the Baker-Gammel-Wills conjecture on the existence of infinite subsequences of diagonal approximants to meromorphic functions outside a closed disk, converging locally uniformly on sets omitting the poles. In view of the length of the paper it is not possible to cover in detail the wealth of material presented. The main subject is the study of hyperelliptic functions, analytic at infinity, with \(2m\) branch points (\(m\geq 1\)), which can be represented as \[ f(z)=r_1(z)+r_2(z)y\text{ with }y:=\sqrt{(z-a_1)\cdots(z-a_{2m})}, \] where \(r_1, r_2\) are rational functions. The main results proved in the paper concern a.o. the `recapturing' of poles \(z_0\) of \(f\) by \([n/n]\) PA's with \(n\geq n_0\), when there are so-called `spurious poles' (say \(k\)), the PA has \(k\) zeros that converge to \(z_0\), an upper bound on the number of spurious poles, an upper bound for the distance between spurious poles and the accompanying zeros from item 2, existence of subsequences of diagonal PA's having prescribed spurious poles, locally uniform convergence after `pole-clearing', proof of the Baker-Gammel-Wills conjecture. The final section gives a new proof of a result due to \textit{S. Dumas} [Sur le développement des fonctions elliptiques en fractions continues, Thésis, Univ. Zürich, (1908; FJM 39.05170.01)] on the hyperelliptic function \[ f(z)=\sqrt{(z-a_1)(_z-a_2)(z-a_3)(z-a_4)} -z^2+(a_1+a_2+a_3+a_4)z/2, \] which is analytic at infinity, the \(a_j\) are pairwise distinct.
scientific article; zbMATH DE number 1013859
scientific article