{"paper":{"title":"Wiener's problem for positive definite functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dmitry Gorbachev, Sergey Tikhonov","submitted_at":"2016-04-05T15:36:24Z","abstract_excerpt":"We study the sharp constant $W_{n}(D)$ in Wiener's inequality for positive definite functions \\[ \\int_{\\mathbb{T}^{n}}|f|^{2}\\,dx\\le W_{n}(D)|D|^{-1}\\int_{D}|f|^{2}\\,dx,\\quad D\\subset \\mathbb{T}^{n}. \\] N. Wiener proved that $W_{1}([-\\delta,\\delta])<\\infty$, $\\delta\\in (0,1/2)$. E. Hlawka showed that $W_{n}(D)\\le 2^{n}$, where $D$ is an origin-symmetric convex body.\n  We sharpen Hlawka's estimates for $D$ being the ball $B^{n}$ and the cube $I^{n}$. In particular, we prove that $W_{n}(B^{n})\\le 2^{(0.401\\ldots +o(1))n}$. We also obtain a lower bound of $W_{n}(D)$. Moreover, for a cube $ D=\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.01302","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"}