{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:BGKX7E7HFM7SJYLEXCZA4VWA5R","short_pith_number":"pith:BGKX7E7H","schema_version":"1.0","canonical_sha256":"09957f93e72b3f24e164b8b20e56c0ec73d161cccf7f0b3f24c68133f50f7d24","source":{"kind":"arxiv","id":"1711.03614","version":2},"attestation_state":"computed","paper":{"title":"Realizations and Factorizations of Positive Definite Kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.SP","authors_text":"Feng Tian, Palle Jorgensen","submitted_at":"2017-11-09T21:47:36Z","abstract_excerpt":"Given a fixed sigma-finite measure space $\\left(X,\\mathscr{B},\\nu\\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $\\nu$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\\left(\\cdot,\\cdot\\right)$ with the two variables from the subset of the sigma-algebra $\\mathscr{B}$, sets having finite $\\nu$ measure. Our setting and results are motivated by applications. The latter are co"},"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":"1711.03614","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2017-11-09T21:47:36Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"7b7b4c130c8e57476d07b0085713a9ff1fd2fefe8cd1b83118fe8cb24fa0af9b","abstract_canon_sha256":"5c12d16931b3c6f97f54aa59775304d8f36cffc4aa5e274fb933f503e0bbbf44"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:03.300193Z","signature_b64":"Jv394dQJc281B1DY/cUPoaVYl4vb2UP5wJDHqUW8BSirGX/sXHHvw5cqqcYf1gtV42HOKLCazOQJTZcq9ipBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09957f93e72b3f24e164b8b20e56c0ec73d161cccf7f0b3f24c68133f50f7d24","last_reissued_at":"2026-05-18T00:02:03.299578Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:03.299578Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Realizations and Factorizations of Positive Definite Kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.SP","authors_text":"Feng Tian, Palle Jorgensen","submitted_at":"2017-11-09T21:47:36Z","abstract_excerpt":"Given a fixed sigma-finite measure space $\\left(X,\\mathscr{B},\\nu\\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $\\nu$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\\left(\\cdot,\\cdot\\right)$ with the two variables from the subset of the sigma-algebra $\\mathscr{B}$, sets having finite $\\nu$ measure. Our setting and results are motivated by applications. The latter are co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03614","kind":"arxiv","version":2},"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":"1711.03614","created_at":"2026-05-18T00:02:03.299679+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.03614v2","created_at":"2026-05-18T00:02:03.299679+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.03614","created_at":"2026-05-18T00:02:03.299679+00:00"},{"alias_kind":"pith_short_12","alias_value":"BGKX7E7HFM7S","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BGKX7E7HFM7SJYLE","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BGKX7E7H","created_at":"2026-05-18T12:31:08.081275+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/BGKX7E7HFM7SJYLEXCZA4VWA5R","json":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R.json","graph_json":"https://pith.science/api/pith-number/BGKX7E7HFM7SJYLEXCZA4VWA5R/graph.json","events_json":"https://pith.science/api/pith-number/BGKX7E7HFM7SJYLEXCZA4VWA5R/events.json","paper":"https://pith.science/paper/BGKX7E7H"},"agent_actions":{"view_html":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R","download_json":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R.json","view_paper":"https://pith.science/paper/BGKX7E7H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.03614&json=true","fetch_graph":"https://pith.science/api/pith-number/BGKX7E7HFM7SJYLEXCZA4VWA5R/graph.json","fetch_events":"https://pith.science/api/pith-number/BGKX7E7HFM7SJYLEXCZA4VWA5R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R/action/storage_attestation","attest_author":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R/action/author_attestation","sign_citation":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R/action/citation_signature","submit_replication":"https://pith.science/pith/BGKX7E7HFM7SJYLEXCZA4VWA5R/action/replication_record"}},"created_at":"2026-05-18T00:02:03.299679+00:00","updated_at":"2026-05-18T00:02:03.299679+00:00"}