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On a density problem related to a theorem of Szegő

Chiara Paulsen

Abstract

A classical theorem of Szegő states that for any probability measure $μ=w\frac{\mathrm{d}θ}{2π}+μ_s$ on the unit circle the polynomials are dense in $L^2(\mathbb{T},μ)$ if and only if $\log(w)\notin L^1(\mathbb{T})$. A related question asks whether the monomials with exponents in some subset $Λ\subseteq \mathbb{N}_0$ already span $L^2(\mathbb{T},μ)$ if $\log(w)\notin L^1(\mathbb{T})$. A result by Olevskii and Ulanovskii gives an answer if $μ$ belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.

On a density problem related to a theorem of Szegő

Abstract

A classical theorem of Szegő states that for any probability measure on the unit circle the polynomials are dense in if and only if . A related question asks whether the monomials with exponents in some subset already span if . A result by Olevskii and Ulanovskii gives an answer if belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.

Paper Structure

This paper contains 4 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Facts from OPUC
  3. Main Theorem
  4. Applications

Key Result

Theorem 1

Let $\Gamma\subseteq \mathbb{N}$ with Then $(\mathbb{N}_0\backslash \Gamma,\mathcal{W}\cap Sz^c)\in\mathscr{A}$.

Theorems & Definitions (30)

  • Remark 1.1
  • Theorem : Olevskii, Ulanovskii
  • Theorem 1.2
  • Theorem 2.1: Szegő
  • Corollary 2.2
  • Corollary 2.3
  • Remark 2.4
  • Lemma 2.5: Properties of Markoff measures
  • proof
  • Remark 2.6
  • ...and 20 more