Everywhere divergence of one-sided ergodic Hilbert transform

Abstract

For a given number α ϵ (0, 1) and a 1-periodic function f, we study the convergence of the series Σ∞ n=1f(x+nα)/n, called one-sided Hilbert transform relative to the rotation x → x + α mod 1. Among others, we prove that for any non-polynomial function of class C2 having Taylor-Fourier series (i.e. Fourier coefficients vanish on ℤ-), there exists an irrational number α (actually a residual set of α) such that the series diverges for all x. We also prove that for any irrational number α, there exists a continuous function f such that the series diverges for all x. The convergence of general series Σ∞ n=1 anf(x + nα) is also discussed in different cases involving the diophantine property of the number α and the regularity of the function f.