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Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals

Klaus Schiefermayr, Olivier Sète

Abstract

We consider Walsh's conformal map from the complement of a compact set $E = \cup_{j=1}^\ell E_j$ with $\ell$ components onto a lemniscatic domain $\widehat{\mathbb{C}} \setminus L$, where $L$ has the form $L = \{ w \in \mathbb{C} : \prod_{j=1}^\ell \lvert w - a_j \rvert^{m_j} \leq \operatorname{cap}(E) \}$. We prove that the exponents $m_j$ appearing in $L$ satisfy $m_j = μ_E(E_j)$, where $μ_E$ is the equilibrium measure of $E$. When $E$ is the union of $\ell$ real intervals, we derive a fast algorithm for computing the centers $a_1, \ldots, a_\ell$. For $\ell = 2$, the formulas for $m_1, m_2$ and $a_1, a_2$ are explicit. Moreover, we obtain the conformal map numerically. Our approach relies on the real and complex Green's functions of $\widehat{\mathbb{C}} \setminus E$ and $\widehat{\mathbb{C}} \setminus L$.

Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals

Abstract

We consider Walsh's conformal map from the complement of a compact set with components onto a lemniscatic domain , where has the form . We prove that the exponents appearing in satisfy , where is the equilibrium measure of . When is the union of real intervals, we derive a fast algorithm for computing the centers . For , the formulas for and are explicit. Moreover, we obtain the conformal map numerically. Our approach relies on the real and complex Green's functions of and .
Paper Structure (6 sections, 14 theorems, 85 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 6 sections, 14 theorems, 85 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Exponents in Terms of the Equilibrium Measure
  3. The Lemniscatic Domain for General Sets
  4. The Lemniscatic Domain for Several Intervals
  5. The Conformal Map for Several Intervals
  6. Proofs

Key Result

Theorem 1.1

Let $E_1, \ldots, E_\ell \subseteq \mathbb{C}$ be disjoint simply connected, infinite compact sets and let that is, $E^{\operatorname{c}} \mathrel{\mathop:}= \widehat{\mathbb{C}} \setminus E$ is an $\ell$-connected domain. Then there exists a unique compact set $L$ of the form where $a_1,\ldots,a_\ell\in\mathbb{C}$ are pairwise distinct and $m_1, \ldots, m_\ell > 0$ are real with $\sum_{j=1}^\el

Figures (5)

  • Figure 1: Error curves $\lvert a_j^{[k]} - a_j\rvert$ in Algorithm \ref{['algo:aj']} applied to the sets $E$ in Example \ref{['ex:analytic']} (ii) (left) and (iii) (right). Missing dots mean that the error is exactly zero.
  • Figure 2: Illustration of the proof of Theorem \ref{['thm:Phi_eqn_unique_sol']}\ref{['it:Phi_eqn_unique_real']}: The graph of the function $F_2(w) = g_L(w)$ for $w \in \mathbb{R} \setminus L$.
  • Figure 3: Left: Set $E = [-1, -0.3] \cup [0.1, 1]$ in Example \ref{['ex:two_intervals_continued']} with a grid. Right: $\partial L$ (green curves), $a_1, a_2$ (black dots) and the image of the grid under $\Phi$.
  • Figure 4: Left: Set $E = [-2, -0.9] \cup [-0.7, 0.2] \cup [0.5, 2.2]$ in Example \ref{['ex:three_intervals']} with a grid. Right: $\partial L$ (green curves), $a_1, a_2, a_3$ (black dots) and the image of the grid under $\Phi$.
  • Figure 5: Panels 1 and 3: Sets $E_2$ and $E_3$ from the construction of the Cantor middle third set (see Example \ref{['ex:cantor']}) with a grid. Panels 2 and 4: Corresponding sets $L$ (with $\partial L$ in green), centers $a_j$ (black dots), and image of the grid under $\Phi$.

Theorems & Definitions (39)

  • Theorem 1.1: Walsh1956
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 29 more