Everywhere divergence of the one-sided ergodic Hilbert transform

Fractal Geometry and Dynamics

27 September 14:00 - 14:50

Jörg Schmeling - Lund University

For a given number $\alpha\in (0, 1)$ and a 1-periodic function $f$, we study the convergence of the series $\sum_{n=1}^\infty\frac{f(x+n\alpha)}{n}$ , called one-sided Hilbert transform relative to the rotation $x \to x + \alpha$ mod 1. Among others, we prove that for any non-polynomial function of class $C^2$ having Taylor-Fourier series (i.e. Fourier coefficients vanish on $\mathbb{Z}_-$), there exists an irrational number $\alpha$ (actually a residual set of $\alpha$'s) such that the series diverges for {\bf all} $x$. We also prove that for {\bf any} irrational number $\alpha$, there exists a continuous function $f$ such that the series diverges for {\bf all} x. The convergence of general series $\sum_{n=1}^\infty a_nf(x + n\alpha)$ is also discussed in different cases involving the diophantine property of the number $\alpha$ and the regularity of the function $f$. Joint work with Aihua Fan.
Kenneth J. Falconer
University of St Andrews
Maarit Järvenpää
University of Oulu
Antti Kupiainen
University of Helsinki
Francois Ledrappier
University of Notre Dame
Pertti Mattila
University of Helsinki


Maarit Järvenpää


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