Everywhere divergence of one-sided ergodic Hilbert transform

Fan, Aihua; Schmeling, Jörg

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2477-2500.

Etant donné un nombre $\alpha \in (0,1)$ et une fonction $1$-périodique $f$, nous étudions la convergence de la série ${\sum }_{n=1}^{\infty }\frac{f(x+n\alpha )}{n}$, appelée la transformée de Hilbert latérale relative à la rotation $x\mapsto x+\alpha mod1$. 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 ${\mathbb{Z}}_{-}$), il existe un $\alpha$ irrationnel (en réalité, un ensemble de $\alpha$ de deuxième catégorie au sens de Baire) tel que la série diverge pour tous les $x$. Nous démontrons aussi que pour tout $\alpha$ irrationnel, il existe une fonction continue $f$ telle que la série diverge pour tous les $x$. La convergence d’une série générale ${\sum }_{n=1}^{\infty }{a}_{n}f(x+n\alpha )$ est aussi discutée pour divers cas où interviennent la propriété diophantienne du nombre $\alpha$ et la régularité de la fonction $f$.

For a given number $\alpha \in (0,1)$ and a $1$-periodic function $f$, we study the convergence of the series ${\sum }_{n=1}^{\infty }\frac{f(x+n\alpha )}{n}$, called one-sided Hilbert transform relative to the rotation $x\mapsto x+\alpha mod1$. Among others, we prove that for any non-polynomial function of class $C^2$ having Taylor–Fourier series (i.e. Fourier coefficients vanish on ${\mathbb{Z}}_{-}$), there exists an irrational number $\alpha$ (actually a residual set of $\alpha$) such that the series diverges for all $x$. We also prove that for any irrational number $\alpha$, there exists a continuous function $f$ such that the series diverges for all $x$. The convergence of general series ${\sum }_{n=1}^{\infty }{a}_{n}f(x+n\alpha )$ is also discussed in different cases involving the diophantine property of the number $\alpha$ and the regularity of the function $f$.

Classification : 37A30, 37A45
Keywords: Ergodic Hilbert transform, Everywhere divergence, Irrational rotation
Mot clés : Transformée de Hilbert ergodique, Divergence partout, Rotation irrationelle

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