Local L 2 -regularity of Riemann’s Fourier series
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 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 : 2014-05-05
Revised : 2016-10-11
Accepted : 2016-10-27
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3135
Classification:  42A20,  11K60,  28C15,  28A78
Keywords: Fourier series, Diophantine approximation, local regularity, Hausdorff dimension
@article{AIF_2017__67_5_2237_0,
     author = {Seuret, St\'ephane and Ubis, Adri\'an},
     title = {Local $L^2$-regularity  of Riemann's Fourier series},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2237-2264},
     doi = {10.5802/aif.3135},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_5_2237_0}
}
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. https://aif.centre-mersenne.org/item/AIF_2017__67_5_2237_0/

[1] Calderón, Alberto P.; Zygmund, Antoni Local properties of solutions of elliptic partial differential equations, Stud. Math., Tome 20 (1961), pp. 171-227 | Article

[2] Chamizo, Fernando; Ubis, Adrián Some Fourier Series with gaps, J. Anal. Math., Tome 101 (2007), pp. 179-197 | Article

[3] Chamizo, Fernando; Ubis, Adrián Multifractal behavior of polynomial Fourier series, Advances in Mathematics, Tome 250 (2014), pp. 1-34 | Article

[4] Duistermaat, Johannes J. Selfsimilarity of “Riemann’s nondifferentiable function”, Nieuw Arch. Wiskd., Tome 9 (1991) no. 3, pp. 303-337

[5] Gerver, Joseph L. The differentiability of the Riemann function at certain rational multiples of π, Am. J. Math., Tome 92 (1970), pp. 33-55 | Article

[6] Hardy, Godfrey H. Weierstrass’s non-differentiable function, American M. S. Trans., Tome 17 (1916), pp. 301-325

[7] Hardy, Godfrey H.; Littlewood, John E. Some problems of Diophantine approximation II: The trigonometrical series associated with the elliptic ϑ-functions, Acta Math., Tome 37 (1914), pp. 193-239 | Article

[8] Itatsu, Seiichi Differentiability of Riemann’s function, Proc. Japan Acad. Ser. A Math. Sci., Tome 57 (1981), pp. 492-495 | Article

[9] Jaffard, Stéphane The spectrum of singularities of Riemann’s function, Rev. Mat. Iberoam., Tome 12 (1996) no. 2, pp. 441-460 | Article

[10] Jaffard, Stéphane; Mélot, Clothilde Wavelet analysis of fractal boundaries. I: Local exponents, Commun. Math. Phys., Tome 258 (2005) no. 3, pp. 513-539 | Article

[11] Rivoal, Tanguy; Seuret, Stéphane Hardy-Littlewood Series and even continued fractions, J. Anal. Math., Tome 125 (2015) no. 1, pp. 175-225 | Article

[12] Stein, Elias M. Harmonic Analysis: Real-variable methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton Mathematical Series, Tome 43 (1993), xiii+695 pages