Jörg Schmeling: Everywhere divergence of the one-sided ergodic Hilbert transform

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.