{"paper":{"title":"Approximation Rates for Interpolation of Sobolev Functions via Gaussians and Allied Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Keaton Hamm","submitted_at":"2013-10-10T17:20:59Z","abstract_excerpt":"A \\Riesz-basis sequence for $L_2[-\\pi,\\pi]$ is a strictly increasing sequence $X:=(x_j)_{j\\in\\mathbb{Z}}$ in $\\mathbb{R}$ such that the set of functions $\\left(e^{-ix_j(\\cdot)}\\right)_{j\\in\\mathbb{Z}}$ is a Riesz basis for $L_2[-\\pi,\\pi]$. Given such a sequence and a parameter $0<h\\leq1$, we consider interpolation of functions $g\\in W_2^k(\\mathbb{R})$ at the set $(hx_j)_{j\\in\\mathbb{Z}}$ via translates of the Gaussian kernel. Existence is shown of an interpolant of the form $$I^{hX}(g)(x):=\\underset{j\\in\\mathbb{Z}}{\\sum}a_je^{-(x-hx_j)^2},\\quad x\\in\\mathbb{R},$$ which is continuous and square-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2892","kind":"arxiv","version":3},"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"}