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

TL;DR

This paper develops convergent multipoint Pade interpolation schemes for Cauchy transforms of complex densities with Jacobi-type weights on analytic arcs, using advanced Riemann-Hilbert techniques and $ar ext{d}$-extensions to establish strong asymptotics and convergence.

Contribution

It introduces a novel approach combining Riemann-Hilbert analysis and $ar ext{d}$-extensions to analyze multipoint Pade interpolation on complex arcs with Jacobi weights.

Findings

Established strong asymptotics for denominator polynomials.

Proved convergence of the interpolation schemes.

Extended Riemann-Hilbert techniques to non-analytic settings.

Abstract

We design convergent multipoint Pade 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 our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Pade interpolants, from which convergence follows.