EVERYWHERE DIVERGENCE OF ONE-SIDED ERGODIC HILBERT TRANSFORM
Résumé
For a given number alpha is an element of (0, 1) and a 1-periodic function f, we study the convergence of the series Sigma(infinity)(n=1) f(x+n alpha)/n, called one-sided Hilbert transform relative to the rotation x bar right arrow 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 alpha) 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 Sigma(infinity)(n=1) a(n)f(x + n alpha) is also discussed in different cases involving the diophantine property of the number alpha and the regularity of the function f.