{"paper":{"title":"Approximation by crystal-refinable function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Alejandro Quintero, Maria del Carmen Moure, Ursula Molter","submitted_at":"2017-01-28T00:11:14Z","abstract_excerpt":"Let $\\Gamma$ be a crystal group in $\\mathbb R^d$. A function $\\varphi:\\mathbb R^d\\longrightarrow \\mathbb C$ is said to be {\\em crystal-refinable} (or $\\Gamma-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $\\varphi(\\gamma^{-1}(ax))$, where the {\\em translations} $\\gamma$ are taken on a crystal group $\\Gamma$, and $a$ is an expansive dilation matrix such that $a\\Gamma a^{-1}\\subset\\Gamma.$ A $\\Gamma-$refinable function $\\varphi: \\mathbb R^d \\rightarrow \\mathbb C$ satisfies a refinement equation $\\varphi(x)=\\sum_{\\gamma\\in\\Gamma}d_\\gamma \\varph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08226","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"}