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Everywhere divergence of one-sided ergodic Hilbert transform. (Divergence partout de la transformée de Hilbert ergodique latérale.) (English. French summary) Zbl 1407.37003

Summary: 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 \(x \mapsto x + \alpha \operatorname{mod} 1\). 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 \(f\).

MSC:

37A30 Ergodic theorems, spectral theory, Markov operators
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
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