{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:C2HXG4KUAVNYVHGS3WGSE322U3","short_pith_number":"pith:C2HXG4KU","schema_version":"1.0","canonical_sha256":"168f737154055b8a9cd2dd8d226f5aa6cc6480f7af595c30e00b3ea4ad889cba","source":{"kind":"arxiv","id":"2606.30786","version":1},"attestation_state":"computed","paper":{"title":"Sobolev-Mercer Expansions and Applications to Stochastic Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Daniel Constantin Rademacher","submitted_at":"2026-06-29T18:15:36Z","abstract_excerpt":"We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(\\Theta)$, where $\\Theta \\subset \\mathbb{R}^d$ is a bounded domain and $k\\in\\mathbb{N}_0$ corresponds to the order of weak differentiability. The spectral decomposition of these operators then yields Mercer-type expansions that are optimal in $H^k(\\Theta\\times\\Theta)$. Notably, we derive from the embedding properties of Sobolev spaces, that for $k>d$, these expansions also converge uniformly without requiring the kernel to be positive definite"},"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":"2606.30786","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-29T18:15:36Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"f137d2c516f9c11980a2ee3d58caf31e59998b57e74efcba1ce9aa5de528b5a0","abstract_canon_sha256":"4a8d96c632f8e921d680765b5e191be37cc5fa6b04096a2b104a738b39e864b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T00:17:16.855010Z","signature_b64":"qUB3+8f5gvyrlzjSFrZpXcAEQGfQTu/cKRxVMTRrCdgv4sgoMEfwBT0eaHTdCHf4z70TZH7o4q4vejPuoTnjCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"168f737154055b8a9cd2dd8d226f5aa6cc6480f7af595c30e00b3ea4ad889cba","last_reissued_at":"2026-07-01T00:17:16.854599Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T00:17:16.854599Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sobolev-Mercer Expansions and Applications to Stochastic Processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Daniel Constantin Rademacher","submitted_at":"2026-06-29T18:15:36Z","abstract_excerpt":"We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(\\Theta)$, where $\\Theta \\subset \\mathbb{R}^d$ is a bounded domain and $k\\in\\mathbb{N}_0$ corresponds to the order of weak differentiability. The spectral decomposition of these operators then yields Mercer-type expansions that are optimal in $H^k(\\Theta\\times\\Theta)$. Notably, we derive from the embedding properties of Sobolev spaces, that for $k>d$, these expansions also converge uniformly without requiring the kernel to be positive definite"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30786","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.30786/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.30786","created_at":"2026-07-01T00:17:16.854655+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.30786v1","created_at":"2026-07-01T00:17:16.854655+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.30786","created_at":"2026-07-01T00:17:16.854655+00:00"},{"alias_kind":"pith_short_12","alias_value":"C2HXG4KUAVNY","created_at":"2026-07-01T00:17:16.854655+00:00"},{"alias_kind":"pith_short_16","alias_value":"C2HXG4KUAVNYVHGS","created_at":"2026-07-01T00:17:16.854655+00:00"},{"alias_kind":"pith_short_8","alias_value":"C2HXG4KU","created_at":"2026-07-01T00:17:16.854655+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/C2HXG4KUAVNYVHGS3WGSE322U3","json":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3.json","graph_json":"https://pith.science/api/pith-number/C2HXG4KUAVNYVHGS3WGSE322U3/graph.json","events_json":"https://pith.science/api/pith-number/C2HXG4KUAVNYVHGS3WGSE322U3/events.json","paper":"https://pith.science/paper/C2HXG4KU"},"agent_actions":{"view_html":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3","download_json":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3.json","view_paper":"https://pith.science/paper/C2HXG4KU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.30786&json=true","fetch_graph":"https://pith.science/api/pith-number/C2HXG4KUAVNYVHGS3WGSE322U3/graph.json","fetch_events":"https://pith.science/api/pith-number/C2HXG4KUAVNYVHGS3WGSE322U3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3/action/storage_attestation","attest_author":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3/action/author_attestation","sign_citation":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3/action/citation_signature","submit_replication":"https://pith.science/pith/C2HXG4KUAVNYVHGS3WGSE322U3/action/replication_record"}},"created_at":"2026-07-01T00:17:16.854655+00:00","updated_at":"2026-07-01T00:17:16.854655+00:00"}