Uniform convergence of Padé diagonal approximants for hyperelliptic functions

The uniform convergence of Padé diagonal approximants is studied for functions in some class that is a natural generalization of hyperelliptic functions. The study is based on Nuttall's approach, which consists in the analysis of a certain Riemann boundary-value problem on the corresponding hyperelliptic Riemann surface. In terms of the solution of this problem, a strong asymptotic formula is obtained for non-Hermitian orthogonal polynomials that are the denominators of the Padé approximants. Under some fairly general assumptions, which are formulated in terms of the periods of the complex Green's function corresponding to the problem and which hold in “general position”, a version of the Baker–Gammel–Willes conjecture is proved.