{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:2QFNIWEKOQ7ZEWEX2CSK4CHEHO","short_pith_number":"pith:2QFNIWEK","schema_version":"1.0","canonical_sha256":"d40ad4588a743f925897d0a4ae08e43b9d482e0ca7e75a6c07edc0ed92361640","source":{"kind":"arxiv","id":"1203.6147","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1203.6147","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-03-28T03:44:23Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"d4ca86c99ab97d2f9731e0790727173283b23917a00cfe206050ee24d3e7de38","abstract_canon_sha256":"247a9c87d4c6c694fa523b5fd5cf784f9eb80e36d9718053dac5bb5046d7caf6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:04.048078Z","signature_b64":"qQJ50ReVKLr/Ck4RUc05ujgsOqxqzEjNbKSnAMYAW23KXmyfX0sWRHKviGoBInQMyvz+kKg4XwF+/TaPUImSDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d40ad4588a743f925897d0a4ae08e43b9d482e0ca7e75a6c07edc0ed92361640","last_reissued_at":"2026-05-18T03:59:04.047297Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:04.047297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $1< p'<\\infty$. 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