{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:WIUEGXUIJ2KAYP26SEERNDNLI7","short_pith_number":"pith:WIUEGXUI","schema_version":"1.0","canonical_sha256":"b228435e884e940c3f5e9109168dab47f878e0f378c9f9203e5dd662b946c688","source":{"kind":"arxiv","id":"1109.0109","version":1},"attestation_state":"computed","paper":{"title":"Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Qi Ye","submitted_at":"2011-09-01T07:29:16Z","abstract_excerpt":"In this paper we introduce a generalization of the classical $\\Leb_2(\\Rd)$-based Sobolev spaces with the help of a vector differential operator $\\mathbf{P}$ which consists of finitely or countably many differential operators $P_n$ which themselves are linear combinations of distributional derivatives. We find that certain proper full-space Green functions $G$ with respect to $L=\\mathbf{P}^{\\ast T}\\mathbf{P}$ are positive definite functions. Here we ensure that the vector distributional adjoint operator $\\mathbf{P}^{\\ast}$ of $\\mathbf{P}$ is well-defined in the distributional sense. We then pro"},"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":"1109.0109","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-01T07:29:16Z","cross_cats_sorted":[],"title_canon_sha256":"43b1aba63c0627043644d8b340d1f2939bea095b2e0c68ffdc416032d36b6ec9","abstract_canon_sha256":"dee6e5acaae1fd690c4496c2ab5cf40e3239da253f9f31547542306b62ef64b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:17.392720Z","signature_b64":"Vv1q/LxvqRwZKaDzRfSJZwyqsAzSdRe65KtauA1sjl6HvghgKG9OZSMO9kvhndYZYOquizs3/cPVmrfM4Hu7Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b228435e884e940c3f5e9109168dab47f878e0f378c9f9203e5dd662b946c688","last_reissued_at":"2026-05-18T04:14:17.391985Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:17.391985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Qi Ye","submitted_at":"2011-09-01T07:29:16Z","abstract_excerpt":"In this paper we introduce a generalization of the classical $\\Leb_2(\\Rd)$-based Sobolev spaces with the help of a vector differential operator $\\mathbf{P}$ which consists of finitely or countably many differential operators $P_n$ which themselves are linear combinations of distributional derivatives. We find that certain proper full-space Green functions $G$ with respect to $L=\\mathbf{P}^{\\ast T}\\mathbf{P}$ are positive definite functions. Here we ensure that the vector distributional adjoint operator $\\mathbf{P}^{\\ast}$ of $\\mathbf{P}$ is well-defined in the distributional sense. We then pro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0109","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1109.0109","created_at":"2026-05-18T04:14:17.392098+00:00"},{"alias_kind":"arxiv_version","alias_value":"1109.0109v1","created_at":"2026-05-18T04:14:17.392098+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0109","created_at":"2026-05-18T04:14:17.392098+00:00"},{"alias_kind":"pith_short_12","alias_value":"WIUEGXUIJ2KA","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"WIUEGXUIJ2KAYP26","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"WIUEGXUI","created_at":"2026-05-18T12:26:44.992195+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/WIUEGXUIJ2KAYP26SEERNDNLI7","json":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7.json","graph_json":"https://pith.science/api/pith-number/WIUEGXUIJ2KAYP26SEERNDNLI7/graph.json","events_json":"https://pith.science/api/pith-number/WIUEGXUIJ2KAYP26SEERNDNLI7/events.json","paper":"https://pith.science/paper/WIUEGXUI"},"agent_actions":{"view_html":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7","download_json":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7.json","view_paper":"https://pith.science/paper/WIUEGXUI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1109.0109&json=true","fetch_graph":"https://pith.science/api/pith-number/WIUEGXUIJ2KAYP26SEERNDNLI7/graph.json","fetch_events":"https://pith.science/api/pith-number/WIUEGXUIJ2KAYP26SEERNDNLI7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/action/storage_attestation","attest_author":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/action/author_attestation","sign_citation":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/action/citation_signature","submit_replication":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/action/replication_record"}},"created_at":"2026-05-18T04:14:17.392098+00:00","updated_at":"2026-05-18T04:14:17.392098+00:00"}