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

Nicolas Chevallier (1) , Guy Cohen (2) , Jean-Pierre Conze (3)

1 IRIMAS - Institut de Recherche en Informatique Mathématiques Automatique Signal
2 Department of Electrical and Computer Engineering [Beer-Sheva]
3 IRMAR - Institut de Recherche Mathématique de Rennes

Sufficient conditions have been given for the convergence in norm and a.e. of the ergodic Hilbert transform (\cite{G96}, \cite{CL}, \cite{Cu09}). Here we apply these conditions to the rotated ergodic Hilbert transform $\sum_{n=1}^\infty \frac{\lambda^n}{n} T^nf$, where $\lambda$ 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 $\lambda$'s for which this series does not converge is at most 2 and give examples where this bound is attained.

Mots clés: contractions; one-sided rotated ergodic Hilbert transform

Domaines: Théorie spectrale [math.SP]

Nicolas Chevallier, Guy Cohen, Jean-Pierre Conze. On the convergence of the rotated one-sided ergodic Hilbert transform. Positivity, 2011, 15 (2), pp.253-270. ⟨10.1007/s11117-010-0070-z⟩. ⟨hal-00456567⟩