## 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 language | English |
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Pages (from-to) | 253-270 |

Number of pages | 18 |

Journal | Positivity |

Volume | 15 |

Issue number | 2 |

DOIs | |

State | Published - 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