Everywhere divergence of one-sided ergodic Hilbert transform
Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2477-2500.

For a given number α(0,1) and a 1-periodic function f, we study the convergence of the series n=1 f(x+nα) n, called one-sided Hilbert transform relative to the rotation xx+αmod1. Among others, we prove that for any non-polynomial function of class C 2 having Taylor–Fourier series (i.e. Fourier coefficients vanish on - ), 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 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.

Etant donné un nombre α(0,1) et une fonction 1-périodique f, nous étudions la convergence de la série n=1 f(x+nα) n, appelée la transformée de Hilbert latérale relative à la rotation xx+αmod1. Entre autres, nous démontrons que pour toute fonction non-polynomiale de classe C 2 admettant une série de Taylor–Fourier (i.e. les coefficients de Fourier sont nuls sur - ), il existe un α irrationnel (en réalité, un ensemble de α de deuxième catégorie au sens de Baire) tel que la série diverge pour tous les x. Nous démontrons aussi que pour tout α irrationnel, il existe une fonction continue f telle que la série diverge pour tous les x. La convergence d’une série générale n=1 a n f(x+nα) est aussi discutée pour divers cas où interviennent la propriété diophantienne du nombre α et la régularité de la fonction f.

Published online:
DOI: 10.5802/aif.3214
Classification: 37A30, 37A45
Keywords: Ergodic Hilbert transform, Everywhere divergence, Irrational rotation
Mot clés : Transformée de Hilbert ergodique, Divergence partout, Rotation irrationelle
Fan, Aihua 1; Schmeling, Jörg 2

1 Central China Normal University School of Mathematics and Statistics 152 Luoyu Road Wuhan 430079 (PRC)
2 Lund University Centre for Mathematical Sciences Box 118, 221 00 LUND (Sweden)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Fan, Aihua and Schmeling, J\"org},
     title = {Everywhere divergence of one-sided ergodic {Hilbert} transform},
     journal = {Annales de l'Institut Fourier},
     pages = {2477--2500},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3214},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3214/}
AU  - Fan, Aihua
AU  - Schmeling, Jörg
TI  - Everywhere divergence of one-sided ergodic Hilbert transform
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2477
EP  - 2500
VL  - 68
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3214/
DO  - 10.5802/aif.3214
LA  - en
ID  - AIF_2018__68_6_2477_0
ER  - 
%0 Journal Article
%A Fan, Aihua
%A Schmeling, Jörg
%T Everywhere divergence of one-sided ergodic Hilbert transform
%J Annales de l'Institut Fourier
%D 2018
%P 2477-2500
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3214/
%R 10.5802/aif.3214
%G en
%F AIF_2018__68_6_2477_0
Fan, Aihua; Schmeling, Jörg. Everywhere divergence of one-sided ergodic Hilbert transform. Annales de l'Institut Fourier, Volume 68 (2018) no. 6, pp. 2477-2500. doi : 10.5802/aif.3214. https://aif.centre-mersenne.org/articles/10.5802/aif.3214/

[1] Assani, Idris; Lin, Michael On the one-sided ergodic Hilbert transform, Ergodic theory and related fields (Contemporary Mathematics), Volume 430, American Mathematical Society, 2007, pp. 21-39 | DOI | MR

[2] Chevallier, Nicolas; Cohen, Guy; Conze, Jean-Pierre On the convergence of the rotated one-sided ergodic Hilbert transform, Positivity, Volume 15 (2011) no. 2, pp. 253-270 | DOI | MR

[3] Cohen, Guy; Lin, Michael The one-sided ergodic Hilbert transform of normal contractions, Characteristic functions, scattering functions and transfer functions (Operator Theory: Advances and Applications), Volume 197, Birkhäuser, 2010, pp. 77-98 | MR

[4] Cuny, Christophe On the a.s. convergence of the one-sided ergodic Hilbert transform, Ergodic Theory Dyn. Syst., Volume 29 (2009) no. 6, pp. 1781-1788 | DOI | MR

[5] Cuny, Christophe Pointwise ergodic theorems with rate with applications to limit theorems for stationary processes, Stoch. Dyn., Volume 11 (2011) no. 1, pp. 135-155 | DOI | MR

[6] Cuny, Christophe; Fan, Aihua Study of almost everywhere convergence of series by mean of martingale methods, Stochastic Processes Appl., Volume 127 (2017) no. 8, pp. 2725-2750 | DOI | MR

[7] Cuny, Christophe; Weber, Michel Ergodic theorems with arithmetical weights, Isr. J. Math., Volume 217 (2017) no. 1, pp. 139-180 | DOI | MR

[8] Dowker, Yael Naim; Erdős, Paul Some examples in ergodic theory, Proc. Lond. Math. Soc., Volume 9 (1959), pp. 227-241 | DOI | MR

[9] Fan, Aihua Almost everywhere convergence of ergodic series, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 2, pp. 490-511 | DOI | MR

[10] Gaposhkin, V. F. On series with respect to the system {ϕ(nx)}, Mat. Sb. (N.S.), Volume 69 (111) (1966), pp. 328-353 | MR | Zbl

[11] Gaposhkin, V. F. Spectral criteria for the existence of generalized ergodic transformations, Teor. Veroyatn. Primen., Volume 41 (1996) no. 2, pp. 251-271 | DOI | MR | Zbl

[12] Iwaniec, Henryk On the problem of Jacobsthal, Demonstr. Math., Volume 11 (1978) no. 1, pp. 225-231 | MR | Zbl

[13] Izumi, Shin-ichi A non-homogeneous ergodic theorem, Proc. Imp. Acad., Tokyo, Volume 15 (1939), pp. 189-192 http://projecteuclid.org/euclid.pja/1195579344 | MR | Zbl

[14] Jacobsthal, Ernst Über Sequenzen ganzer Zahlen, von denen keine zu n teilerfremd ist. I, II, III, Norske Vid. Selsk. Forhdl., Volume 33 (1961), p. 117-124, 125–131, 132–139 | MR | Zbl

[15] del Junco, Andrés; Rosenblatt, Joseph Counterexamples in ergodic theory and number theory, Math. Ann., Volume 245 (1979) no. 3, pp. 185-197 | DOI | MR

[16] Kahane, Jean-Pierre Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, 50, Springer, 1970, viii+169 pages | MR | Zbl

[17] Khinchin Continued fractions, University of Chicago Press, 1964, xi+95 pages | MR | Zbl

Cited by Sources: