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Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Vasiliy A. Prokhorov

Abstract

Let $ρ_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on a compact set $E\subset \mathbb{C}$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$, respectively. Motivated by Hadamard's classical theorem on Hankel determinants and by Gonchar's theorem on rows of the Walsh table, we study, for each fixed $m\ge 0$, the asymptotic behavior as $n\to\infty$ of the products $$ \prod_{k=0}^{m}ρ_{n-m+k,k}(f;E). $$ We establish Hadamard-type asymptotic formulas for these products on the closed unit disc and, more generally, on continua with connected complement and Jordan boundary. In the disc case, our approach combines Hadamard's classical theorem and Gonchar's theorem with weighted Hankel operators and an AAK-type theorem for meromorphic approximation. We also show that there exists a common subsequence along which the extremal exponential behavior of these products and of the corresponding products on the closed Green sublevel sets $E_R$ is attained.

Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Abstract

Let denote the error of best uniform rational approximation to a function analytic on a compact set by rational functions whose numerator and denominator have degrees at most and , respectively. Motivated by Hadamard's classical theorem on Hankel determinants and by Gonchar's theorem on rows of the Walsh table, we study, for each fixed , the asymptotic behavior as of the products We establish Hadamard-type asymptotic formulas for these products on the closed unit disc and, more generally, on continua with connected complement and Jordan boundary. In the disc case, our approach combines Hadamard's classical theorem and Gonchar's theorem with weighted Hankel operators and an AAK-type theorem for meromorphic approximation. We also show that there exists a common subsequence along which the extremal exponential behavior of these products and of the corresponding products on the closed Green sublevel sets is attained.

Paper Structure

This paper contains 21 sections, 11 theorems, 272 equations.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Main results
  4. Hadamard's Theorem, Hankel Determinants, and AAK Theory
  5. Power series, Hankel determinants, and Hadamard's theorem
  6. Hankel operators, bilinear forms, and AAK-type estimates on domains with analytic boundary
  7. The Disc Case
  8. Passage to the exterior problem and the disc AAK setup
  9. The determinantal upper bound and the proof of Proposition \ref{['prop:disc_product']}
  10. Proof of Theorem \ref{['thm:two']} in the disc case
  11. Proof of Theorem \ref{['thm:hankel-disc-quotient-R']}
  12. Comparison of products on two circles
  13. Proof of Theorem \ref{['thm:first-main']} in the disc case
  14. From the disc to continua with analytic boundary and the Jordan-boundary case
  15. Conformal reduction and Faber expansion
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $E\subset\mathbb{C}$ be a continuum with at least two points and connected complement, and let $f$ be analytic on $E$. Fix $m\ge0$, and assume that $f$ is not a rational function with at most $m$ finite poles, counted with multiplicity. Let $R$ satisfy $1<R<R_0$. Then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 6 more