Everywhere divergence of onesided ergodic hilbert transform
(2018) In Annales de l'Institut Fourier 68(6). p.24772500 Abstract
For a given number α ϵ (0, 1) and a 1periodic function f, we study the convergence of the series Σ^{∞}
_{n=1}f(x+nα)/n, called onesided Hilbert transform relative to the rotation x → x + α mod 1. Among others, we prove that for any nonpolynomial function of class C^{2} having TaylorFourier 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 1periodic function f, we study the convergence of the series Σ^{∞}
(Less)
_{n=1}f(x+nα)/n, called onesided Hilbert transform relative to the rotation x → x + α mod 1. Among others, we prove that for any nonpolynomial function of class C^{2} having TaylorFourier 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.
 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
 03730956
 language
 English
 LU publication?
 yes
 id
 867f50860c7243cbbcc8f28290078c21
 date added to LUP
 20190107 13:54:30
 date last changed
 20210929 01:15:38
@article{867f50860c7243cbbcc8f28290078c21, abstract = {<p>For a given number α ϵ (0, 1) and a 1periodic function f, we study the convergence of the series Σ<sup>∞</sup><br> <sub>n=1</sub>f(x+nα)/n, called onesided Hilbert transform relative to the rotation x → x + α mod 1. Among others, we prove that for any nonpolynomial function of class C<sup>2</sup> having TaylorFourier 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 = {03730956}, language = {eng}, number = {6}, pages = {24772500}, publisher = {ANNALES DE L INSTITUT FOURIER}, series = {Annales de l'Institut Fourier}, title = {Everywhere divergence of onesided ergodic hilbert transform}, volume = {68}, year = {2018}, }