{"paper":{"title":"Upper Beurling Density of Systems formed by Translates of finite Sets of Elements in $L^p(\\R^d)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Bei Liu, Rui Liu","submitted_at":"2012-03-28T03:44:23Z","abstract_excerpt":"In this paper, we prove that if a finite disjoint union of translates $\\bigcup_{k=1}^n\\{f_k(x-\\gamma)\\}_{\\gamma\\in\\Gamma_k}$ in $L^p(\\R^d)$ $(1<p<\\infty)$ is a $p'$-Bessel sequence for some $1<p'<\\infty$, then the disjoint union $\\Gamma=\\bigcup_{k=1}^n\\Gamma_k$ has finite upper Beurling density, and that if $\\bigcup_{k=1}^n\\{f_k(x-\\gamma)\\}_{\\gamma\\in\\Gamma_k}$ is a $(C_q)$-system with $1/p+1/q=1$, then $\\Gamma$ has infinite upper Beurling density. Thus, no finite disjoint union of translates in $L^p(\\R^d)$ can form a $p'$-Bessel $(C_q)$-system for any $1< p'<\\infty$. Furthermore, by using tec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6147","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"}