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 \(\sum_{n=1}^\infty f(x+n )/n \), called one-sided Hilbert transform relative to the rotation \(x \mapsto x+\alpha \mod 1\). Among others, we prove that for any non-polynomial function of class \(C^2 \) having Taylor- Fourier series (i.e. Fourier coefficients vanish on \(Z_- \)), there exists an irrational number alpha (actually a residual set of alphas) such that the series diverges for all x. We also prove that for any irrational number alpha , there exists a continuous function f such that the series diverges for all x. The convergence of general series \(\sum_{n=1}^\infty a_nf(x+n)\) is also discussed in different cases involving the diophantine property of the number alpha and the regularity of the function f.