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

Cade Ballew, Thomas Trogdon

TL;DR

The work develops a fast numerical framework based on the Fokas–Its–Kitaev Riemann–Hilbert representation to compute the inverse spectral map for measures supported on multiple disjoint intervals. By applying lensing, an exterior Green's function via a differential $\mathfrak{g}'$, and a gap-eliminating auxiliary function $\mathfrak{h}_n$, the authors transform the RH problem into a numerically tractable form and extract the first $N$ recurrence coefficients in $O(N)$ arithmetic. The approach bypasses theta-function evaluations, encodes endpoint singularities through weighted Chebyshev Cauchy integrals, and yields fast convergence with applications to Toda lattices and function approximation on disconnected domains. The method offers potential extensions to Gauss quadrature for such weights and may enable efficient computation of Riemann theta functions numerically.

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.

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

TL;DR

The work develops a fast numerical framework based on the Fokas–Its–Kitaev Riemann–Hilbert representation to compute the inverse spectral map for measures supported on multiple disjoint intervals. By applying lensing, an exterior Green's function via a differential , and a gap-eliminating auxiliary function , the authors transform the RH problem into a numerically tractable form and extract the first recurrence coefficients in arithmetic. The approach bypasses theta-function evaluations, encodes endpoint singularities through weighted Chebyshev Cauchy integrals, and yields fast convergence with applications to Toda lattices and function approximation on disconnected domains. The method offers potential extensions to Gauss quadrature for such weights and may enable efficient computation of Riemann theta functions numerically.

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 method to compute the first 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.
Paper Structure (28 sections, 3 theorems, 160 equations, 17 figures)

This paper contains 28 sections, 3 theorems, 160 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Orthogonal polynomials and Cauchy integrals
  3. Orthogonal polynomials
  4. Cauchy integrals
  5. The Riemann--Hilbert problem
  6. Setting up the Riemann--Hilbert Problem
  7. Transforming the Riemann--Hilbert problem
  8. Step 1: Lensing
  9. Step 2: The differential $\mathfrak g'$
  10. Step 3: Removing the jump conditions on the gaps
  11. Recurrence coefficients
  12. Numerical methods
  13. Numerically computing Cauchy integrals
  14. Numerically computing $\mathfrak{g}$
  15. Asymptotic behavior
  16. ...and 13 more sections

Key Result

Lemma 3.1

The $(g+1)\times(g+1)$ matrix $\mathbf{H}$ defined by is invertible.

Figures (17)

  • Figure 1: A weight function of the form \ref{['weight']} with support on $[-3.2, -2.2]\cup[0.1, 1.1]\cup[2, 3]\cup[3.5, 4]$ with $\alpha_1=\beta_1=-1$ (Chebyshev-$\mathcal{T}$), $\alpha_2=\beta_2=1$ (Chebyshev-$U$), $\alpha_3=1,\beta_3=-1$ (Chebyshev-$V$), $\alpha_4=-1,\beta_4=1$ (Chebyshev-$W$).
  • Figure 2: Sample regions $\Omega_j$ and $D_j$ and curves $C_j$ for a three interval case.
  • Figure 3: Jump conditions in a three interval case. $F^k_j(z)$ are as defined in \ref{['shorthand']}.
  • Figure 4: Plots of the difference between the first 51 orthogonal polynomial coefficients of a Chebyshev-$U$ weight modified with \ref{['hexample']} and an unmodified Chebyshev-$U$ weight.
  • Figure 5: Absolute error in the computed first 51 orthogonal polynomial coefficients for a Chebyshev-$U$ weight modified with \ref{['hexample']} for 2, 4, 8, and 16 collocation points per interval (PPI) on $[-1,1]$.
  • ...and 12 more figures

Theorems & Definitions (5)

  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 4.2: Plemelj Lemma, Muskhelishvili1977, Section 17