Uniform convergence of Padé diagonal approximants for hyperelliptic functions (Q2710726)

In this interesting paper the uniform convergence of Padé diagonal approximants for hyperelliptic functions is investigated. The aim is to prove some results for these approximants of meromorphic functions of the form \(f(z) = \widehat{\mu}(z) + r(z)\) where \(\widehat{\mu}\) is an arbitrary Markov function and \(r(z)\) is a rational function holomorphic on the smallest closed interval \([a,b]\) containing the support of the measure \(\mu\) associated with \(\widehat{\mu}(z)\). Firstly, the problem is studied in the ``Rakhmanov case'' when \(r\in\mathbb{C}(z)\), the support of \(\mu\) is the union of finitely many intervals and the measure \(\mu\) itself is represented by a smooth weight on these intervals in a certain ``general position'' (Theorem 1). Next, the authors investigate also the ``Nuttall case'' (Theorem 2) when the end-points of the support of \(\mu\) lie in the complex plane (here the measure \(\mu\) must be represented by a smooth complex-value weight). The author characterizes the asymptotic behaviour of the Padé approximants, both in the Rakhmanov case and in the Nuttall case, in terms of the solution of a certain Riemann boundary-value problem on the extended complex plane cut along several analytic arcs making up a compact subset \(S\) of minimum capacity (Nuttall approach). A version of the Baker-Gammel-Willes conjecture is finally proved.