# A Riemann--Hilbert approach to computing the inverse spectral map for measures supported on disjoint intervals

## TL;DR

This paper introduces a fast numerical method based on Riemann--Hilbert problems for computing orthogonal polynomials on disjoint intervals, enabling efficient calculation of recurrence coefficients and polynomial evaluations.

## Contribution

It develops an $ ext{O}(N)$ Riemann--Hilbert based approach for orthogonal polynomials on multiple intervals, addressing a gap where analytical formulas are unknown.

## Key findings

- Method outperforms existing techniques in efficiency.
- Demonstrates fast convergence in applications.
- Enables pointwise evaluation of polynomials and Cauchy transforms.

## Abstract

We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\mathrm{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.