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Chebyshev polynomials in the complex plane and on the real line

Olof Rubin

Abstract

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all polynomials with a prescribed leading coefficient, they minimize the supremum norm on a given compact set. Although we do not present new results, we provide -- in selected cases -- new proofs of known theorems and compile a collection of open problems.

Chebyshev polynomials in the complex plane and on the real line

Abstract

We present a survey of central developments in the theory of Chebyshev polynomials, introduced by P.~L.~Chebyshev and later extended to the complex plane by G.~Faber. Our primary focus is their defining extremal property: among all polynomials with a prescribed leading coefficient, they minimize the supremum norm on a given compact set. Although we do not present new results, we provide -- in selected cases -- new proofs of known theorems and compile a collection of open problems.

Paper Structure

This paper contains 14 sections, 48 theorems, 409 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fundamental properties
  3. An extension of the concept, existence and uniqueness
  4. Orthogonality and alternation
  5. Weighted Chebyshev polynomials on an interval
  6. Chebyshev polynomials in the complex plane
  7. Chebyshev polynomials relative to a Jordan curve
  8. Chebyshev polynomials relative to a Jordan arc
  9. Potential theory and Widom factors
  10. Totik--Widom bounds
  11. Chebyshev and Faber polynomials relative to equipotential curves
  12. The zeros of Chebyshev polynomials
  13. Computing Chebyshev polynomials
  14. Open problems

Key Result

Lemma 2.1

Assume that $\mathsf{E} \subset \mathbb{C}$ is a compact set, and let $w : \mathsf{E} \to [0, \infty)$ be an upper semicontinuous function that is positive on at least distinct $n+1$ points. Let $a_0^\ast, \dotsc, a_{n-1}^\ast \in \mathbb{C}$ be such that then there are (at least) $n+1$ distinct points $z_1,\dotsc,z_{n+1}$ in $\mathsf{E}$ such that

Figures (2)

  • Figure 1: Zeros of Chebyshev polynomials corresponding to $m$-cusped Hypoycloids
  • Figure 2: Zeros of Chebyshev polynomials corresponding to the Bernoulli lemniscate $\{z:|z^2-1|=1\}$

Theorems & Definitions (77)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4: Rivlin & Shapiro rivlin-shapiro61
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:rivlin_shapiro']}
  • Theorem 2.6: Borel (1905) borel05, Markov markov48
  • ...and 67 more