{"paper":{"title":"Polynomial Approximation in $ L^2 $ of the Double Exponential via Complex Analysis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Boaz Klartag, Pierre Bizeul","submitted_at":"2025-02-11T10:50:36Z","abstract_excerpt":"We study the polynomial approximation problem in $L^2(\\mu_1)$ where $\\mu_1(dx) = e^{-|x|}/2 dx$. We show that for any absolutely continuous function $f$, $$ \\sum_{k=1}^{\\infty} \\log^2(e+k) \\langle f, P_k \\rangle^2 \\ \\leq C \\left( \\int_{\\mathbb{R}} \\log^2(e+\\lvert x \\rvert) f^2 \\, d\\mu_1 \\ + \\ \\int_{\\mathbb{R}} (f')^2 \\, d\\mu_1 \\right) $$\n  for some universal constant $C>0$, where $(P_k)_{k \\in N}$ are the orthonormal polynomials associated with $\\mu_1$. This inequality is tight in the sense that $\\log^2(e +k)$ on the left hand-side cannot be replaced by $a_k \\log^2(e +k)$ with a sequence $a_k "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.07448","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/2502.07448/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"}