{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:DATZFS7IMJEY2HWESP4SMENE33","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":"1a35c0d132d68e2f6f5ad9163215879a8a5d29ea3087e1108b61b12a7d9d819f","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2009-09-09T20:59:17Z","title_canon_sha256":"1dcc42ed4c0761685ef4b5937cb99911b6378811d5f058b2e5800fd0d2e744e2"},"schema_version":"1.0","source":{"id":"0909.1828","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.1828","created_at":"2026-05-18T03:34:36Z"},{"alias_kind":"arxiv_version","alias_value":"0909.1828v1","created_at":"2026-05-18T03:34:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.1828","created_at":"2026-05-18T03:34:36Z"},{"alias_kind":"pith_short_12","alias_value":"DATZFS7IMJEY","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"DATZFS7IMJEY2HWE","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"DATZFS7I","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:778996a4d94033e2373daa48a910abf2aa43e1152379dc218044068f0637ebd4","target":"graph","created_at":"2026-05-18T03:34:36Z","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":"A certain kernel (sometimes called the Pick kernel) associated to Schur functions on the disk is always positive semi-definite. A generalization of this fact is well-known for Schur functions on the polydisk. In this article, we show that the Pick kernel on the polydisk has a great deal of structure beyond being positive semi-definite. It can always be split into two kernels possessing certain shift invariance properties.","authors_text":"Greg Knese","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2009-09-09T20:59:17Z","title":"Kernel decompositions for Schur functions on the polydisk"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1828","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:ec0e37f0bf6612ce76b6b7a7e32aaedb7247cadd7208ed321cbd2e331b5c84c5","target":"record","created_at":"2026-05-18T03:34:36Z","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":"1a35c0d132d68e2f6f5ad9163215879a8a5d29ea3087e1108b61b12a7d9d819f","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2009-09-09T20:59:17Z","title_canon_sha256":"1dcc42ed4c0761685ef4b5937cb99911b6378811d5f058b2e5800fd0d2e744e2"},"schema_version":"1.0","source":{"id":"0909.1828","kind":"arxiv","version":1}},"canonical_sha256":"182792cbe862498d1ec493f92611a4dee9475370435f8036afba874c9f719d5a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"182792cbe862498d1ec493f92611a4dee9475370435f8036afba874c9f719d5a","first_computed_at":"2026-05-18T03:34:36.084677Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:36.084677Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OhFzYuoQ+wbbixfNX/chxD0bCPAuVclaMeMgZea+E4Vsh6caM4F3j5qFrqjpDh3aWmjWzhKGHeHz1XTRb/HfCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:36.085128Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.1828","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec0e37f0bf6612ce76b6b7a7e32aaedb7247cadd7208ed321cbd2e331b5c84c5","sha256:778996a4d94033e2373daa48a910abf2aa43e1152379dc218044068f0637ebd4"],"state_sha256":"8c7b56799358b82239410b4f135e15abfc2daeda7f74e2c04c3b108fc9e5ba55"}