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Polynomial and rational interpolation: potential, barycentric weights, and Lebesgue constants

Kelong Zhao, Shuhuang Xiang

TL;DR

This work develops a potential-theoretic framework for polynomial and rational interpolation on $[-1,1]$, linking barycentric weights and Lebesgue constants to a continuous potential $U$ and its discrete counterpart $U_n$. It proves that non-constant potentials induce exponential growth in weight ratios and Lebesgue constants, while equilibrium (constant) potentials yield non-exponential upper bounds, and it analyzes how pole placement can enforce equilibrium-like behavior. The Floater-Hormann scheme and pole-based rational interpolation are examined to illustrate when exponential convergence is possible or inhibited, with guidance on constructing external fields to stabilize growth. Overall, the paper provides a unified view connecting node distributions, external fields, and potential convergence to explain interpolation stability and convergence rates in both polynomial and rational contexts.

Abstract

In this paper, we focus on barycentric weights and Lebesgue constants for Lagrange interpolation of arbitrary node distributions on \([-1,1]\). The following three main works are included: estimates of upper and lower bounds on the barycentric weights are given in terms of the logarithmic potential function; for interpolation of non-equilibrium potentials, lower bounds with exponentially growing parts of Lebesgue constants are given; and for interpolation consistent with equilibrium potentials, non-exponentially growing upper bounds on their Lebesgue constants are given. Based on the work in this paper, we can discuss the behavior of the Lebesgue constant and the existence of exponential convergence in a unified manner in the framework of potential theory.

Polynomial and rational interpolation: potential, barycentric weights, and Lebesgue constants

TL;DR

This work develops a potential-theoretic framework for polynomial and rational interpolation on , linking barycentric weights and Lebesgue constants to a continuous potential and its discrete counterpart . It proves that non-constant potentials induce exponential growth in weight ratios and Lebesgue constants, while equilibrium (constant) potentials yield non-exponential upper bounds, and it analyzes how pole placement can enforce equilibrium-like behavior. The Floater-Hormann scheme and pole-based rational interpolation are examined to illustrate when exponential convergence is possible or inhibited, with guidance on constructing external fields to stabilize growth. Overall, the paper provides a unified view connecting node distributions, external fields, and potential convergence to explain interpolation stability and convergence rates in both polynomial and rational contexts.

Abstract

In this paper, we focus on barycentric weights and Lebesgue constants for Lagrange interpolation of arbitrary node distributions on . The following three main works are included: estimates of upper and lower bounds on the barycentric weights are given in terms of the logarithmic potential function; for interpolation of non-equilibrium potentials, lower bounds with exponentially growing parts of Lebesgue constants are given; and for interpolation consistent with equilibrium potentials, non-exponentially growing upper bounds on their Lebesgue constants are given. Based on the work in this paper, we can discuss the behavior of the Lebesgue constant and the existence of exponential convergence in a unified manner in the framework of potential theory.
Paper Structure (9 sections, 8 theorems, 49 equations, 10 figures)

This paper contains 9 sections, 8 theorems, 49 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Barycentric weights and potential
  4. Lebesgue constants for exponential growth
  5. The Lebesgue constant in an equilibrium potential
  6. Interpolation Error and Lebesgue Constant
  7. Take the Floater-Hormann interpolation as an example
  8. Fast and stable rational interpolation
  9. Conclusions

Key Result

Lemma 3.1

\newlabelle:3.10 Assume a class of rational interpolants $r_{n,m}$ with nodes $\{x_i^{(n)}\}_{i=0}^n$ in the interval $[-1,1]$ obeying a positive density function $w$, and whose poles $\{p_j^{(n)}\}_{j=1}^m$ lie outside $[-1,1]$, generating an external field $\phi_n$ that converges to $\phi$. If $

Figures (10)

  • Figure 1: Three potential functions, denoted as $U_a,\,U_b,\,U_c$, along with the differences between their respective maxima and minima: $d_1$, $d_2$, $d_3$. (b): The maximum value of the ratio of absolute values of barycentric weights (top) and the Lebesgue constant (bottom) for the three interpolants. The reference growth rate is $O([\exp d_k]^n)$ ($k=1,2,3$).
  • Figure 1: (a): Lebesgue functions of EXAMPLE 1 at different $n$. (b): Values of Lebesgue functions of EXAMPLE 1 at different points.
  • Figure 1: (a): Schematic representation of \ref{['le:5.1']}. (b): Schematic representation of the proof of \ref{['le:5.2']}.
  • Figure 1: Contours of the potential function $U_n$ generated by $(n+1)$ nodes (blue) and $m$ poles (red). As $n$ grows, the potential function stabilizes in some neighborhood of $[-1,1]$. The computation of these nodes and poles is detailed in Zhao2023.
  • Figure 1: The main research path of this paper.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Lemma 3.1
  • Proof 1
  • Remark 3.2
  • Definition 3.3
  • Lemma 3.4
  • Proof 2
  • Theorem 3.5
  • Proof 3
  • Corollary 3.6
  • ...and 7 more