{"paper":{"title":"Sampling on Paley-Wiener spaces on graphs, with particular focus on the infinite-dimensional case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Sampling sets for Paley-Wiener spaces on graphs are exactly the complements of lambda-sets.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Filippo Giannoni","submitted_at":"2025-11-21T16:04:53Z","abstract_excerpt":"We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where a Poincar\\'e-type inequality holds), thereby providing a sufficient condition for stable sampling and reconstruction on graphs such as $\\mathbb{Z}^n$-lattices and radial trees with finite geometry."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where a Poincaré-type inequality holds), thereby providing a sufficient condition for stable sampling and reconstruction on graphs such as Z^n-lattices and radial trees with finite geometry.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The graphs admit a well-defined infinite-dimensional Paley-Wiener space in which the Poincaré-type inequality on lambda-sets is both necessary and sufficient for the sampling theorem to hold.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Sampling sets for infinite-dimensional Paley-Wiener spaces on graphs are exactly the complements of lambda-sets where a Poincaré inequality holds, enabling stable reconstruction.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sampling sets for Paley-Wiener spaces on graphs are exactly the complements of lambda-sets.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2e49a3938e2616da33af37d89cb8f1f94e7f82535ca9b81eb556965e4e2be1c9"},"source":{"id":"2511.17343","kind":"arxiv","version":6},"verdict":{"id":"3f467035-3acc-45f5-9304-509b7423e8e4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-17T20:10:16.349519Z","strongest_claim":"We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where a Poincaré-type inequality holds), thereby providing a sufficient condition for stable sampling and reconstruction on graphs such as Z^n-lattices and radial trees with finite geometry.","one_line_summary":"Sampling sets for infinite-dimensional Paley-Wiener spaces on graphs are exactly the complements of lambda-sets where a Poincaré inequality holds, enabling stable reconstruction.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The graphs admit a well-defined infinite-dimensional Paley-Wiener space in which the Poincaré-type inequality on lambda-sets is both necessary and sufficient for the sampling theorem to hold.","pith_extraction_headline":"Sampling sets for Paley-Wiener spaces on graphs are exactly the complements of lambda-sets."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2511.17343/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}