Convergent Interpolation to Cauchy Integrals over Analytic Arcs

TL;DR

This paper proves the convergence of multipoint Padé approximants to Cauchy transforms over analytic Jordan arcs, using orthogonal polynomial analysis and explicit interpolation schemes based on arc parametrization.

Contribution

It establishes convergence results for multipoint Padé approximants on analytic arcs with explicit interpolation scheme construction, extending classical approximation theory.

Findings

Padé approximants converge locally uniformly outside the arc

Convergence is linked to properties of non-Hermitian orthogonal polynomials

Explicit interpolation schemes are constructed based on arc parametrization

Abstract

We consider multipoint Pad\'e approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth non-vanishing density, then the diagonal multipoint Pad\'e approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Pad\'e approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the arc.