{"paper":{"title":"The sharp Remez-type inequality for even trigonometric polynomials on the period","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Tam\\'as Erd\\'elyi","submitted_at":"2018-09-20T04:08:24Z","abstract_excerpt":"We prove that $$\\max_{t \\in [-\\pi,\\pi]}{|Q(t)|} \\leq T_{2n}(\\sec(s/4)) = \\frac 12 ((\\sec(s/4) + \\tan(s/4))^{2n} + (\\sec(s/4) - \\tan(s/4))^{2n})$$ for every even trigonometric polynomial $Q$ of degree at most $n$ with complex coefficients satisfying $$m(\\{t \\in [-\\pi,\\pi]: |Q(t)| \\leq 1\\}) \\geq 2\\pi-s\\,, \\qquad s \\in (0,2\\pi)\\,,$$ where $m(A)$ denotes the Lebesgue measure of a measurable set $A \\subset {\\Bbb R}$ and $T_{2n}$ is the Chebysev polynomial of degree $2n$ on $[-1,1]$ defined by $T_{2n}(\\cos t) = \\cos(2nt)$ for $t \\in {\\Bbb R}$. This inequality is sharp. We also prove that $$\\max_{t \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07466","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"}