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Transfinite diameter with generalized polynomial degree

Sione Ma`u

Abstract

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula, we prove that in the unweighted case, these generalized Leja points recover the transfinite diameter.

Transfinite diameter with generalized polynomial degree

Abstract

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula, we prove that in the unweighted case, these generalized Leja points recover the transfinite diameter.

Paper Structure

This paper contains 5 sections, 21 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Chebyshev constants from a submultiplicative function
  3. Chebyshev constants compatible with generalized degree
  4. Vandermonde determinant estimates
  5. $C$-Leja points

Key Result

Theorem 1

Let $K\subset{\mathbb C}^N$ be compact and $w\colon K\to[0,\infty)$ an admissible weight. Then we have the formula

Theorems & Definitions (44)

  • Theorem : Theorem \ref{['thm:Ok']}
  • Definition \oldthetheorem
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  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
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  • Definition \oldthetheorem
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  • ...and 34 more