Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weighted residual polynomials on a circular arc

Jacob S. Christiansen, Benjamin Eichinger, Olof Rubin, Maxim Zinchenko

Abstract

We study the behavior of weighted residual polynomials on circular arcs, including weighted Chebyshev polynomials. For weights given by reciprocals of polynomials, we establish Szegő-Widom asymptotics. Extending our analysis to less regular weights, we determine the asymptotic behavior of the corresponding weighted Widom factors, generalizing results by Eichinger and Thiran et al. As an application, we derive the asymptotics of Widom factors on certain lemniscatic arcs.

Weighted residual polynomials on a circular arc

Abstract

We study the behavior of weighted residual polynomials on circular arcs, including weighted Chebyshev polynomials. For weights given by reciprocals of polynomials, we establish Szegő-Widom asymptotics. Extending our analysis to less regular weights, we determine the asymptotic behavior of the corresponding weighted Widom factors, generalizing results by Eichinger and Thiran et al. As an application, we derive the asymptotics of Widom factors on certain lemniscatic arcs.
Paper Structure (11 sections, 14 theorems, 187 equations, 6 figures)

This paper contains 11 sections, 14 theorems, 187 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Chebyshev polynomials on lemniscatic arcs
  4. Main Result
  5. Proof of Theorem \ref{['thm:Widom_factors_lemniscatic_arcs']}
  6. Analysis of the Limit Points
  7. Reciprocals of polynomials as weight functions
  8. Residual polynomials relative to the origin
  9. Residual polynomials relative to an arbitrary point
  10. General Weight functions
  11. Allowing for zeros

Key Result

Theorem 2.1

Let ${\mathsf E}_{m,r}(\alpha)=\{z: z^m+1\in r \Gamma_\alpha\}$, where $m\in {\mathbb N}$, $r>0$, and $0<\alpha<\pi$. Then, for any $0\leq l <m$, we have where In particular, for the connected case where $r = 1$, we have

Figures (6)

  • Figure 1: A circular arc $\Gamma_\alpha = \{e^{it}: -\alpha\leq t \leq \alpha\}$.
  • Figure 2: Examples of sets in the families ${\mathsf E}_{2,r}(\pi/7)$, ${\mathsf E}_{5,r}(\pi/7)$ and ${\mathsf E}_{7,r}(\pi/7)$ for varying values of $r$. The lines corresponding to $r=1.1$ are dashed.
  • Figure 3: The domain $\Omega_n$ with designated points $\overline{z}_j$.
  • Figure 4: The transformation of the domain $\Omega_\alpha$ (top left) to $\Omega_{\tilde{\alpha}}$ (bottom). The mapping $\varphi_1$ sends $u_0$ to the origin.
  • Figure 5: The Möbius transformation $\Phi$ mapping the auxiliary parameter plane (left) to the original parameter plane (right). The map is determined by the correspondence of the points $0 \mapsto x_0$ and $\zeta_0 \mapsto z_{u_0}$.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 19 more