On the convergence of the rotated one-sided ergodic Hilbert transform

Nicolas Chevallier, Guy Cohen, Jean Pierre Conze

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Sufficient conditions have been given for the convergence in norm and a. e. of the ergodic Hilbert transform (Gaposhkin in Theory Probab Appl 41:247-264, 1996; Cohen and Lin in Characteristic functions, scattering functions and transfer functions, pp 77-98, Birkhäuser, Basel, 2009; Cuny in Ergod Theory Dyn Syst 29:1781-1788, 2009). Here we apply these conditions to the rotated ergodic Hilbert transform, where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ's for which this series does not converge is at most 2 and give examples where this bound is attained.

Original languageEnglish
Pages (from-to)253-270
Number of pages18
JournalPositivity
Volume15
Issue number2
DOIs
StatePublished - 1 Jun 2011

Keywords

  • Contractions
  • Hausdorff dimension
  • One-sided rotated ergodic Hilbert transform
  • Spectral measure

ASJC Scopus subject areas

  • Analysis
  • Theoretical Computer Science
  • General Mathematics

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