Local L²-regularity of Riemann's Fourier series

2014/05/05 by Stéphane Seuret, Seuret, Stéphane, Adrián Ubis +1
Mathematics · #11K60 #28C15. Secondary: 28A78 #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #Analytic Number Theory Research #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Primary: 42A20 #math.FA #math.MG #msc:11K60 #msc:28A78 #msc:28C15. #msc:42A20

paper · pdf · doi:10.48550/arxiv.1405.0810

21 pages, 1 figure

arxiv created 2014/05/05 · openalex publication_date 2014/05/05 · arxiv updated 2014/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We are interested in the convergence and the local regularity of the lacunary Fourier series F_s(x) = ∑_n=1^+∞ \frace^{2iπn2 x}n^s. In the 1850's, Riemann introduced the series F₂ 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<s≤ 1, and we prove that F_s(x) converges when x satisfies a Diophantine condition. We also study the L²- local regularity of F_s, proving that the local L²-norm of F_s around a point x behave differently around different x, according again to Diophantine conditions on x.

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