{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:WIUEGXUIJ2KAYP26SEERNDNLI7","short_pith_number":"pith:WIUEGXUI","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"},"canonical_sha256":"b228435e884e940c3f5e9109168dab47f878e0f378c9f9203e5dd662b946c688","source":{"kind":"arxiv","id":"1109.0109","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.0109","created_at":"2026-05-18T04:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1109.0109v1","created_at":"2026-05-18T04:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0109","created_at":"2026-05-18T04:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"WIUEGXUIJ2KA","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WIUEGXUIJ2KAYP26","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WIUEGXUI","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:WIUEGXUIJ2KAYP26SEERNDNLI7","target":"record","payload":{"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"},"canonical_sha256":"b228435e884e940c3f5e9109168dab47f878e0f378c9f9203e5dd662b946c688","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"},"source_kind":"arxiv","source_id":"1109.0109","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"InAXAU6h0xaqtSscfboRLA8rFl+cEZeBfzHja/tNLApD++fUyorOyb1C/6l15KM4vbIoZwapAV2Od5/a01pIAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T21:36:29.988184Z"},"content_sha256":"574743ede2e84c59e7e5e7fe220c0682e6e605c9a7a768507dc1fb1bdf5b11b4","schema_version":"1.0","event_id":"sha256:574743ede2e84c59e7e5e7fe220c0682e6e605c9a7a768507dc1fb1bdf5b11b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:WIUEGXUIJ2KAYP26SEERNDNLI7","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vDqEBJVibKJaEb372ZPHKJSE/2aQWKZ7tOJGCizP1z0G1nTHzVx5hhokv+ew8Hbrw50DipfLf/ncft8cxPzsAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T21:36:29.988885Z"},"content_sha256":"035c830a8ce3beb718d6df557968d4f55fc071bf18fee616944744021b06d8bb","schema_version":"1.0","event_id":"sha256:035c830a8ce3beb718d6df557968d4f55fc071bf18fee616944744021b06d8bb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/bundle.json","state_url":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T21:36:29Z","links":{"resolver":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7","bundle":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/bundle.json","state":"https://pith.science/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WIUEGXUIJ2KAYP26SEERNDNLI7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:WIUEGXUIJ2KAYP26SEERNDNLI7","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"dee6e5acaae1fd690c4496c2ab5cf40e3239da253f9f31547542306b62ef64b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-01T07:29:16Z","title_canon_sha256":"43b1aba63c0627043644d8b340d1f2939bea095b2e0c68ffdc416032d36b6ec9"},"schema_version":"1.0","source":{"id":"1109.0109","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.0109","created_at":"2026-05-18T04:14:17Z"},{"alias_kind":"arxiv_version","alias_value":"1109.0109v1","created_at":"2026-05-18T04:14:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.0109","created_at":"2026-05-18T04:14:17Z"},{"alias_kind":"pith_short_12","alias_value":"WIUEGXUIJ2KA","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"WIUEGXUIJ2KAYP26","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"WIUEGXUI","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:035c830a8ce3beb718d6df557968d4f55fc071bf18fee616944744021b06d8bb","target":"graph","created_at":"2026-05-18T04:14:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Qi Ye","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-01T07:29:16Z","title":"Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0109","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:574743ede2e84c59e7e5e7fe220c0682e6e605c9a7a768507dc1fb1bdf5b11b4","target":"record","created_at":"2026-05-18T04:14:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"dee6e5acaae1fd690c4496c2ab5cf40e3239da253f9f31547542306b62ef64b3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-09-01T07:29:16Z","title_canon_sha256":"43b1aba63c0627043644d8b340d1f2939bea095b2e0c68ffdc416032d36b6ec9"},"schema_version":"1.0","source":{"id":"1109.0109","kind":"arxiv","version":1}},"canonical_sha256":"b228435e884e940c3f5e9109168dab47f878e0f378c9f9203e5dd662b946c688","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b228435e884e940c3f5e9109168dab47f878e0f378c9f9203e5dd662b946c688","first_computed_at":"2026-05-18T04:14:17.391985Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:17.391985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Vv1q/LxvqRwZKaDzRfSJZwyqsAzSdRe65KtauA1sjl6HvghgKG9OZSMO9kvhndYZYOquizs3/cPVmrfM4Hu7Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:17.392720Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.0109","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:574743ede2e84c59e7e5e7fe220c0682e6e605c9a7a768507dc1fb1bdf5b11b4","sha256:035c830a8ce3beb718d6df557968d4f55fc071bf18fee616944744021b06d8bb"],"state_sha256":"bcd7853651dd0c1be784817d7dd8613426312863b994ede0ff7f1adae031b865"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FBSYWqhxjmabYMBTSYwvL6ru9e+bWaN2iQaBWLSswOWy4GwA4OUs5P7G5lLrFK/rtCRTdPIKxWXTeRrA+ZN4Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T21:36:29.991689Z","bundle_sha256":"0210b5fb9ae58e742e00854f33c0f0d29b50d19891de3a93a9b0326ceaf92c16"}}