{"paper":{"title":"Local $L^2$-regularity of Riemann's Fourier series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.FA","authors_text":"Adri\\'an Ubis, St\\'ephane Seuret","submitted_at":"2014-05-05T07:47:08Z","abstract_excerpt":"We are interested in the convergence and the local regularity of the lacunary Fourier series $F_s(x) = \\sum_{n=1}^{+\\infty} \\frac{e^{2i\\pi n^2 x}}{n^s}$. 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 $1/2<s\\leq 1$, 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 $L^2$-norm of $F_s$ around a point $x$ behave diffe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0810","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}