Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II

TL;DR

This work advances the practical computation of Walsh's conformal map from the exterior of a polynomial pre-image $E=P^{-1}([-1,1])$ onto a lemniscatic domain $L$, by determining the centers $a_j$ and exponents $m_j$ that define $L$ through $U(w)=\prod_{j=1}^\ell (w-a_j)^{m_j}$ with $\sum m_j=1$. Building on the result from Part I that $m_j= n_j/n$, where $n_j$ counts zeros of $P$ in $E_j$, the authors develop an iterative method to compute the centers $a_j$ for general $\ell$ real-interval configurations, and provide explicit center formulas in the two- and three-component symmetric cases. They also present a Kammerer-type algorithm for determining $a_j$ when $E$ comprises an arbitrary number of intervals, together with numerical validation and cross-checks against boundary-integral methods, illustrating high accuracy and fast convergence. The results enable accurate construction of Walsh maps and related Faber–Walsh polynomials for multiply connected sets, with potential applications in polynomial approximation and potential theory on complex domains.

Abstract

We consider Walsh's conformal map from the exterior of a set $E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint components onto a lemniscatic domain. In particular, we are interested in the case when $E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where $P$ is an algebraic polynomial of degree $n$. Of special interest are the exponents and the centers of the lemniscatic domain. In the first part of this series of papers, a very simple formula for the exponents has been derived. In this paper, based on general results of the first part, we give an iterative method for computing the centers when $E$ is the union of $\ell$ intervals. Once the centers are known, the corresponding Walsh map can be computed numerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ components satisfying certain symmetry relations then the centers and the corresponding Walsh map are given by explicit formulas. All our theorems are illustrated with analytical or numerical examples.

Figures (10)

• Figure 1: Real graph of a polynomial $P$ of degree $n = 7$ whose pre-image of $[-1, 1]$ consists of $\ell = 4$ intervals; see Example \ref{['ex:deg7_ell4']} for the explicit formula of $P$.
• Figure 2: Illustration of the Walsh map $\Phi$ for $E = [-1, b_2] \cup [b_3, 1]$ with $b_2 = \frac{1}{2} (1 - \alpha^2) - \alpha$ and $b_3 = \frac{1}{2} (1 - \alpha^2) + \alpha$ for $\alpha = 0.1, 0.05, 0.01$ (from top to bottom); see Example \ref{['ex:P3_two_intervals']}. Left: Original domain with intervals (black) and a grid. Right: $\partial L$ (black) and image of the grid under $\Phi$.
• Figure 3: $E = P^{-1}([-1, 1]) = [-2, -0.2] \cup [0.2, 2]$ in Example \ref{['ex:two_sym_real_intervals']}. Left: $E$ (black) and a grid. Right: $\partial L$ (black), $a_1, a_2$ (black dots), and the image of the grid under $\Phi$.
• Figure 4: Pre-image $E = P^{-1}([-1, 1])$ in Example \ref{['ex:intersecting_arcs']} with $\alpha = 1.01$. Left: $E$ (black lines) and a grid. Right: $\partial L$ (black curves), $a_1, a_2$ (black dots), and the image of the grid under $\Phi$.
• Figure 5: Left: The set $E$ in Example \ref{['ex:three_intervals']} with $\alpha = 0.05$ and $\beta = 0.3$ in black and a grid. Right: Corresponding lemniscatic domain ($\partial L$ in black), the points $a_1, a_2, a_3$ (black dots), and the image of the grid under $\Phi$.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5