{"paper":{"title":"A Sharp Fourier Inequality and the Epanechnikov Kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.CA","authors_text":"Sean Richardson","submitted_at":"2023-10-15T02:42:42Z","abstract_excerpt":"We consider functions $f: \\mathbb{Z} \\to \\mathbb{R}$ and kernels $u: \\{-n, \\cdots, n\\} \\to \\mathbb{R}$ normalized by $\\sum_{\\ell = -n}^{n} u(\\ell) = 1$, making the convolution $u \\ast f$ a \"smoother\" local average of $f$. We identify which choice of $u$ most effectively smooths the second derivative in the following sense. For each $u$, basic Fourier analysis implies there is a constant $C(u)$ so $\\|\\Delta(u \\ast f)\\|_{\\ell^2(\\mathbb{Z})} \\leq C(u)\\|f\\|_{\\ell^2(\\mathbb{Z})}$ for all $f: \\mathbb{Z} \\to \\mathbb{R}$. By compactness, there is some $u$ that minimizes $C(u)$ and in this paper, we fi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.09713","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2310.09713/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}