{"paper":{"title":"On the largest critical value of $T_n^{(k)}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alexei Shadrin, Geno Nikolov, Nikola Naidenov","submitted_at":"2017-10-17T06:43:32Z","abstract_excerpt":"We study the quantity $$ \\tau_{n,k}:=\\frac{|T_n^{(k)}(\\omega_{n,k})|}{T_n^{(k)}(1)}\\,, $$ where $T_n$ is the Chebyshev polynomial of degree $n$, and $\\omega_{n,k}$ is the rightmost zero of $T_n^{(k+1)}$.\n  Since the absolute values of the local maxima of $T_n^{(k)}$ increase monotonically towards the end-points of $[-1,1]$, the value $\\tau_{n,k}$ shows how small is the largest critical value of $\\,T_n^{(k)}\\,$ relative to its global maximum $\\,T_n^{(k)}(1)$.\n  In this paper, we improve and extend earlier estimates by Erd\\H{o}s--Szeg\\H{o}, Eriksson and Nikolov in several directions.\n  Firstly, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.06120","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"}