{"paper":{"title":"Optimal-Degree Polynomial Approximations for Exponentials and Gaussian Kernel Density Estimation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.CA"],"primary_cat":"cs.CC","authors_text":"Amol Aggarwal, Josh Alman","submitted_at":"2022-05-12T17:47:29Z","abstract_excerpt":"For any real numbers $B \\ge 1$ and $\\delta \\in (0, 1)$ and function $f: [0, B] \\rightarrow \\mathbb{R}$, let $d_{B; \\delta} (f) \\in \\mathbb{Z}_{> 0}$ denote the minimum degree of a polynomial $p(x)$ satisfying $\\sup_{x \\in [0, B]} \\big| p(x) - f(x) \\big| < \\delta$. In this paper, we provide precise asymptotics for $d_{B; \\delta} (e^{-x})$ and $d_{B; \\delta} (e^{x})$ in terms of both $B$ and $\\delta$, improving both the previously known upper bounds and lower bounds. In particular, we show $$d_{B; \\delta} (e^{-x}) = \\Theta\\left( \\max \\left\\{ \\sqrt{B \\log(\\delta^{-1})}, \\frac{\\log(\\delta^{-1}) }{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.06249","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/2205.06249/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"}