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Everywhere divergence of one-sided ergodic hilbert transform

Fan, Aihua and Schmeling, Jörg LU (2018) In Annales de l'Institut Fourier 68(6). p.2477-2500
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... (More)

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.

(Less)
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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Ergodic Hilbert transform, Everywhere divergence, Irrational rotation
in
Annales de l'Institut Fourier
volume
68
issue
6
pages
24 pages
publisher
Association des Annales de l'Institut Fourier
external identifiers
  • scopus:85057802696
ISSN
0373-0956
language
English
LU publication?
yes
id
867f5086-0c72-43cb-bcc8-f28290078c21
date added to LUP
2019-01-07 13:54:30
date last changed
2025-10-14 09:54:55
@article{867f5086-0c72-43cb-bcc8-f28290078c21,
  abstract     = {{<p>For a given number α ϵ (0, 1) and a 1-periodic function f, we study the convergence of the series Σ<sup>∞</sup><br>
                            <sub>n=1</sub>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<sup>2</sup> having Taylor-Fourier series (i.e. Fourier coefficients vanish on ℤ<sub>-</sub>), 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<sup>∞</sup><br>
                            <sub>n=1</sub> a<sub>n</sub>f(x + nα) is also discussed in different cases involving the diophantine property of the number α and the regularity of the function f.</p>}},
  author       = {{Fan, Aihua and Schmeling, Jörg}},
  issn         = {{0373-0956}},
  keywords     = {{Ergodic Hilbert transform; Everywhere divergence; Irrational rotation}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{2477--2500}},
  publisher    = {{Association des Annales de l'Institut Fourier}},
  series       = {{Annales de l'Institut Fourier}},
  title        = {{Everywhere divergence of one-sided ergodic hilbert transform}},
  volume       = {{68}},
  year         = {{2018}},
}