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On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights

Thomas Trogdon

TL;DR

This work studies the strong asymptotics of orthogonal polynomials on multiple intervals with nonanalytic Jacobi-like weights. It develops a two-stage ∂̄ (dbar) deformation of the Fokas–Its–Kitaev Riemann–Hilbert problem to accommodate differentiable but nonanalytic perturbations, and constructs a global parametrix from Szegő data and Riemann theta functions while treating endpoint neighborhoods with Bessel-type parametrices. The authors derive explicit recurrence-coefficient asymptotics and provide rigorous error bounds that depend on smoothness and endpoint exponents, extending Yattselev’s Chebyshev-like results to the general Jacobi-like case and offering new Chebyshev-like improvements and point-mass extensions. Numerical experiments suggest the current bounds may be suboptimal in some regimes and highlight the potential for further rate enhancements and algorithmic applications to compute recurrence coefficients with near-optimal complexity. Overall, the paper broadens the scope of RH methods for nonanalytic weights on multi-interval supports and lays groundwork for perturbative analyses in random matrices and related areas.

Abstract

We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its $\overline{\partial}$ extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative.

On the Asymptotics of Orthogonal Polynomials on Multiple Intervals with Non-Analytic Weights

TL;DR

This work studies the strong asymptotics of orthogonal polynomials on multiple intervals with nonanalytic Jacobi-like weights. It develops a two-stage ∂̄ (dbar) deformation of the Fokas–Its–Kitaev Riemann–Hilbert problem to accommodate differentiable but nonanalytic perturbations, and constructs a global parametrix from Szegő data and Riemann theta functions while treating endpoint neighborhoods with Bessel-type parametrices. The authors derive explicit recurrence-coefficient asymptotics and provide rigorous error bounds that depend on smoothness and endpoint exponents, extending Yattselev’s Chebyshev-like results to the general Jacobi-like case and offering new Chebyshev-like improvements and point-mass extensions. Numerical experiments suggest the current bounds may be suboptimal in some regimes and highlight the potential for further rate enhancements and algorithmic applications to compute recurrence coefficients with near-optimal complexity. Overall, the paper broadens the scope of RH methods for nonanalytic weights on multi-interval supports and lays groundwork for perturbative analyses in random matrices and related areas.

Abstract

We consider the asymptotics of orthogonal polynomials for measures that are differentiable, but not necessarily analytic, multiplicative perturbations of Jacobi-like measures supported on disjoint intervals. We analyze the Fokas-Its-Kitaev Riemann-Hilbert problem using the Deift-Zhou method of nonlinear steepest descent and its extension due to Miller and McLaughlin. Our results extend that of Yattselev in the case of Chebyshev-like measures with error bounds that give similar rates while allowing less regular perturbations. For the general Jacobi-like case, we present, what appears to be the first result for asymptotics when the perturbation of the measure is only assumed to be differentiable with bounded second derivative.

Paper Structure

This paper contains 29 sections, 16 theorems, 211 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The asymptotics of the recurrence coefficients
  3. The Fokas--Its--Kitaev Riemann--Hilbert problem
  4. Dbar deformation
  5. The first $\boldsymbol{ \overline{\partial}}$ deformation of the FIK RH problem
  6. The second $\boldsymbol{\overline{\partial}}$ deformation of the FIK RH problem
  7. Construction of the global parametrix
  8. Using local solutions
  9. Global approximation
  10. Extensions and improvements
  11. Chebyshev-like polynomials on multiple intervals
  12. The addition of a finite number of point masses
  13. Asymptotics of recurrence coefficients and estimating optimal errors
  14. Conclusions and outlook
  15. The $\boldsymbol{ g}$-function
  16. ...and 14 more sections

Key Result

Lemma 2.1

Figures (6)

  • Figure 1: Model contours. Here we use $\sigma = 2\pi/3$. The contours are symmetric about the real axis.
  • Figure 2: Mapping the contours $\Gamma_j$ using the local conformal mappings, with valid choices for $\theta_1$, $\theta_2$ displayed.
  • Figure 3: The addition of contours to form $\Sigma_{j,\pm}$ and enclose the regions $\Omega_{j,\pm}$. The definition of $\Sigma_{j,\pm}$ includes the mapped contours near $a_j$, $b_j$.
  • Figure 4: A comparison of the recurrence coefficients when $\gamma = 3/2$. The observed discrepancy between the recurrence coefficients and their known asymptotics appears to be $\mathrm{O}\bigl(n^{-\gamma-1}\bigr)$.
  • Figure 5: A comparison of the recurrence coefficients when $\gamma = 2$. The observed discrepancy between the recurrence coefficients and their known asymptotics appears to be $\mathrm{O}\bigl(n^{-\gamma-1}\bigr)$.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 3.1
  • Lemma 3.2: norm bounds
  • ...and 8 more