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

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2010 by Springer Basel AG
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

References

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