Local L 2 -regularity of Riemann’s Fourier series
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2237-2264.

We are interested in the convergence and the local regularity of the lacunary Fourier series F s (x)= n=1 + e 2iπn 2 x n s . In the 1850’s, Riemann introduced the series F 2 as a possible example of nowhere differentiable function, and the study of this function has drawn the interest of many mathematicians since then. We focus on the case when 1/2<s1, and we prove that F s (x) converges when x satisfies a Diophantine condition. We also study the L 2 - local regularity of F s , proving that the local L 2 -norms of F s around a point x behave differently around different x, according again to Diophantine conditions on x.

Dans cet article, nous nous intéressons aux propriétés de convergence et de régularité locale des séries de Fourier lacunaires F s (x)= n=1 + e 2iπn 2 x n s . Dans les années 1850, Riemann avait proposé la série F 2 comme exemple possible de fonction continue nulle part dérivable. La non-dérivabilité de F 2 et plus généralement sa régularité locale ont depuis lors été étudiées par de nombreux mathématiciens, soulevant des questions d’analyse harmonique, d’analyse complexe et d’approximation diophantienne. Nous considérons le cas 1/2<s1, et trouvons un critère diophantien sur x pour la convergence de F s (x). Nous étudions également la régularité locale de F s , en démontrant que les L 2 -exposants de F s dépendent de conditions diophantiennes sur x. Les preuves utilisent des estimées locales sur la norme L 2 des sommes partielles de F s .

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3135
Classification: 42A20, 11K60, 28C15, 28A78
Keywords: Fourier series, Diophantine approximation, local regularity, Hausdorff dimension
Mot clés : Séries de Fourier, Approximation diophantienne, Régularité locale, Dimension de Hausdorff
Seuret, Stéphane 1; Ubis, Adrián 2

1 Université Paris-Est LAMA (UMR 8050) UPEMLV, UPEC, CNRS, 94010, Créteil (France)
2 Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid 28049 Madrid (Spain)
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Seuret, Stéphane; Ubis, Adrián. Local $L^2$-regularity  of Riemann’s Fourier series. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2237-2264. doi : 10.5802/aif.3135. http://www.numdam.org/articles/10.5802/aif.3135/

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