{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:ESG4RM7EBTR54UGFBIV2NVWLWS","short_pith_number":"pith:ESG4RM7E","schema_version":"1.0","canonical_sha256":"248dc8b3e40ce3de50c50a2ba6d6cbb48545fe77b83246e440c127cfa99486b1","source":{"kind":"arxiv","id":"1204.6448","version":3},"attestation_state":"computed","paper":{"title":"Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Gregory E. Fasshauer, Qi Ye","submitted_at":"2012-04-29T02:27:46Z","abstract_excerpt":"In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\\mathbf{P}^{\\ast T}\\mathbf{P}$ now becomes a conditionally pos"},"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":"1204.6448","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2012-04-29T02:27:46Z","cross_cats_sorted":[],"title_canon_sha256":"f8481ed7402b11c7b6d6b1e4510c0d9fb616cd27727e834f0be8c21f85368010","abstract_canon_sha256":"e945f4658c2ee5a23cbc776e73b7e11674283b8aec441a7649b954e3281bf684"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:05.599165Z","signature_b64":"EsOxEev7KvTZBKOWQLHSxSsz8DEXYMEgYUeQvloFObFeMhYyXRUjkeEja2kss2eTPKDOXFzfyAFHQeZInSkFAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"248dc8b3e40ce3de50c50a2ba6d6cbb48545fe77b83246e440c127cfa99486b1","last_reissued_at":"2026-05-18T03:32:05.598695Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:05.598695Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Gregory E. Fasshauer, Qi Ye","submitted_at":"2012-04-29T02:27:46Z","abstract_excerpt":"In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\\mathbf{P}^{\\ast T}\\mathbf{P}$ now becomes a conditionally pos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.6448","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1204.6448","created_at":"2026-05-18T03:32:05.598754+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.6448v3","created_at":"2026-05-18T03:32:05.598754+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.6448","created_at":"2026-05-18T03:32:05.598754+00:00"},{"alias_kind":"pith_short_12","alias_value":"ESG4RM7EBTR5","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"ESG4RM7EBTR54UGF","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"ESG4RM7E","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS","json":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS.json","graph_json":"https://pith.science/api/pith-number/ESG4RM7EBTR54UGFBIV2NVWLWS/graph.json","events_json":"https://pith.science/api/pith-number/ESG4RM7EBTR54UGFBIV2NVWLWS/events.json","paper":"https://pith.science/paper/ESG4RM7E"},"agent_actions":{"view_html":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS","download_json":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS.json","view_paper":"https://pith.science/paper/ESG4RM7E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.6448&json=true","fetch_graph":"https://pith.science/api/pith-number/ESG4RM7EBTR54UGFBIV2NVWLWS/graph.json","fetch_events":"https://pith.science/api/pith-number/ESG4RM7EBTR54UGFBIV2NVWLWS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS/action/storage_attestation","attest_author":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS/action/author_attestation","sign_citation":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS/action/citation_signature","submit_replication":"https://pith.science/pith/ESG4RM7EBTR54UGFBIV2NVWLWS/action/replication_record"}},"created_at":"2026-05-18T03:32:05.598754+00:00","updated_at":"2026-05-18T03:32:05.598754+00:00"}