## Abstract

For a given number α ϵ (0, 1) and a 1-periodic function f, we study the convergence of the series Σ^{∞}
_{n=1}f(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 C^{2} 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} a_{n}f(x + nα) is also discussed in different cases involving the diophantine property of the number α and the regularity of the function f.

Original language | English |
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Pages (from-to) | 2477-2500 |

Number of pages | 24 |

Journal | Annales de l'Institut Fourier |

Volume | 68 |

Issue number | 6 |

Publication status | Published - 2018 |

## Subject classification (UKÄ)

- Mathematical Analysis

## Free keywords

- Ergodic Hilbert transform
- Everywhere divergence
- Irrational rotation