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

On the convergence of the rotated one-sided ergodic Hilbert transform 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 15 (2) – Jun 10, 2010
18 pages

/lp/springer_journal/on-the-convergence-of-the-rotated-one-sided-ergodic-hilbert-transform-NxvRUkhV6H
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0070-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Jun 10, 2010

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