{"paper":{"title":"Sampling Theorems for Shift-invariant Spaces, Gabor Frames, and Totally Positive Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.FA","authors_text":"Joachim St\\\"ockler, Jos\\'e Luis Romero, Karlheinz Gr\\\"ochenig","submitted_at":"2016-12-02T12:18:28Z","abstract_excerpt":"We study nonuniform sampling in shift-invariant spaces and the construction of Gabor frames with respect to the class of totally positive functions whose Fourier transform factors as $ \\hat g(\\xi)= \\prod_{j=1}^n (1+2\\pi i\\delta_j\\xi)^{-1} \\, e^{-c \\xi^2}$ for $\\delta_1,\\ldots,\\delta_n\\in \\mathbb{R}, c >0$ (in which case $g$ is called totally positive of Gaussian type).\n  In analogy to Beurling's sampling theorem for the Paley-Wiener space of entire functions, we prove that every separated set with lower Beurling density $>1$ is a sampling set for the shift-invariant space generated by such a $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00651","kind":"arxiv","version":2},"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"}