Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Padé interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on the work [9] for the choice of the interpolation points and dwell on the Riemann–Hilbert approach to asymptotics of orthogonal polynomials introduced in [33] in the case of a segment. We also elaborate on the $\bar \partial$-extension of the Riemann–Hilbert technique, initiated in [37] on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Padé interpolants, from which convergence follows.