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Positivity

, Volume 15, Issue 2, pp 253–270 | Cite as

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

  • Nicolas Chevallier
  • Guy Cohen
  • Jean-Pierre Conze
Article
  • 63 Downloads

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 \({\sum_{n=1}^\infty \frac{\lambda^n}{n} T^nf}\) , 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.

Keywords

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

Mathematics Subject Classification (2000)

Primary 47A35 47B15 Secondary 37A30 42A16 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Nicolas Chevallier
    • 1
  • Guy Cohen
    • 2
  • Jean-Pierre Conze
    • 3
  1. 1.Faculté des Sciences et TechniquesUniversité de Haute AlsaceMulhouseFrance
  2. 2.Department of Electrical EngineeringBen-Gurion UniversityBeer ShevaIsrael
  3. 3.IRMAR, UMR CNRS 6625Université de Rennes IRennes CedexFrance

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