Everywhere divergence of one-sided ergodic hilbert transform
(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∞
(Less)
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.
- author
- Fan, Aihua and Schmeling, Jörg LU
- organization
- publishing date
- 2018
- 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
- 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
- 2022-03-25 07:13:10
@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 = {{ANNALES DE L INSTITUT FOURIER}}, series = {{Annales de l'Institut Fourier}}, title = {{Everywhere divergence of one-sided ergodic hilbert transform}}, volume = {{68}}, year = {{2018}}, }