{"paper":{"title":"Sharp Remez inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"P. Yuditskii, S. Tikhonov","submitted_at":"2018-09-25T21:11:52Z","abstract_excerpt":"Let an algebraic polynomial $P_n(\\zeta)$ of degree $n$ be such that $|P_n(\\zeta)|\\le 1$ for $\\zeta\\in E\\subset\\mathbb{T}$ and $|E|\\ge 2\\pi -s$. We prove the sharp Remez inequality $$ \\sup_{\\zeta\\in\\mathbb{T}}|P_n(\\zeta)|\\le \\mathfrak{T}_{n}\\left(\\sec \\frac{s} 4\\right),$$ where $\\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$ P_n(e^{iz})=e^{i(nz/2+c_1)}\\mathfrak{T}_n\\left(\\sec\\frac s 4\\cos \\frac {z-c_0} 2\\right), \\quad c_0,c_1\\in\\mathbb{R}. $$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.09726","kind":"arxiv","version":2},"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"}