{"paper":{"title":"On alternative quantization for doubly weighted approximation and integration over unbounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"F. Pillichshammer, G.W. Wasilkowski, L. Plaskota, P. Kritzer","submitted_at":"2019-07-09T07:00:39Z","abstract_excerpt":"It is known that for a $\\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to $\\|\\omega^{1/\\alpha}\\|_{L_1}^\\alpha\\|f^{(r)}\\psi\\|_{L_p}n^{-r+(1/p-1/q)_+},$ where $\\omega=\\rho/\\psi$ and $\\alpha=r-1/p+1/q.$ Moreover, the optimal sample points are determined by quantiles of $\\omega^{1/\\alpha}.$ In this paper, we show how the error of best approximations changes when the sample points are determined by a quantizer $\\kappa$ other than $\\omega.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04015","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"}