{"paper":{"title":"Tractability of the function approximation problem in terms of the kernel's shape and scale parameters","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Fred J. Hickernell, Xuan Zhou","submitted_at":"2014-11-04T05:48:09Z","abstract_excerpt":"This article studies the problem of approximating functions belonging to a Hilbert space $\\mathcal H_d$ with a reproducing kernel of the form $$\\tilde K_d(\\boldsymbol x,\\boldsymbol t):=\\prod_{\\ell=1}^d \\left(1-\\alpha_\\ell^2+\\alpha_\\ell^2K_{\\gamma_\\ell}(x_\\ell,t_\\ell)\\right)\\ \\ \\ \\mbox{for all} \\ \\ \\ \\boldsymbol x,\\boldsymbol t\\in\\mathbb R^d.$$ The $\\alpha_\\ell\\in[0,1]$ are scale parameters, and the $\\gamma_\\ell>0$ are sometimes called shape parameters. The reproducing kernel $K_{\\gamma}$ corresponds to some Hilbert space of functions defined on $\\mathbb R$. The kernel $\\tilde K_d$ generalizes "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0790","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"}