{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:LH62RKRBNW37WDGZPH4FXGQRVB","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":"ec2eb32bb7bfeb2b7b1ac5a8eb831b4963a9f7474b115623e487f3379914cca5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-06-03T15:52:36Z","title_canon_sha256":"535b6099e42ca5cb5c3640e2661606b80c7a7a178640b3beb507a3c52e90fdc6"},"schema_version":"1.0","source":{"id":"1406.0762","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.0762","created_at":"2026-05-18T02:50:32Z"},{"alias_kind":"arxiv_version","alias_value":"1406.0762v1","created_at":"2026-05-18T02:50:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.0762","created_at":"2026-05-18T02:50:32Z"},{"alias_kind":"pith_short_12","alias_value":"LH62RKRBNW37","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"LH62RKRBNW37WDGZ","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"LH62RKRB","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:f7144dc41f6ff916dbcb0ea46abce2ff1a9e04f4048291501fc2257a6cf8d85c","target":"graph","created_at":"2026-05-18T02:50:32Z","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":"Orthogonal polynomials on the product domain $[a_1,b_1] \\times [a_2,b_2]$ with respect to the inner product $$\n  \\langle f,g \\rangle_S = \\int_{a_1}^{b_1} \\int_{a_2}^{b_2} \\nabla f(x,y)\\cdot \\nabla g(x,y)\\, w_1(x)w_2(y)\n  \\,dx\\, dy + \\lambda f(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a weight function on $[a_i,b_i]$ for $i = 1, 2$, $\\lambda > 0$, and $(c_1, c_2)$ is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which ","authors_text":"F. Marcell\\'an, L. Fern\\'andez, M. A. Pi\\~nar, T. E. P\\'erez, Y. Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-06-03T15:52:36Z","title":"Sobolev orthogonal polynomials on product domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0762","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:cb493c516e6451229e757638eb144cec55f43141c97134d048accd62a19a1c58","target":"record","created_at":"2026-05-18T02:50:32Z","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":"ec2eb32bb7bfeb2b7b1ac5a8eb831b4963a9f7474b115623e487f3379914cca5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-06-03T15:52:36Z","title_canon_sha256":"535b6099e42ca5cb5c3640e2661606b80c7a7a178640b3beb507a3c52e90fdc6"},"schema_version":"1.0","source":{"id":"1406.0762","kind":"arxiv","version":1}},"canonical_sha256":"59fda8aa216db7fb0cd979f85b9a11a863f6664596c82680cfba9b1e70f68dfd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"59fda8aa216db7fb0cd979f85b9a11a863f6664596c82680cfba9b1e70f68dfd","first_computed_at":"2026-05-18T02:50:32.883031Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:32.883031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DH6y4vv96Ev5XPcwWnDEipbpp99d/7d8K3sAtcn3Pe9w1Qu9GwVN8p4xOY7fXRtTdPSgPIQrbDB3U1bjDGG8Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:32.883661Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.0762","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cb493c516e6451229e757638eb144cec55f43141c97134d048accd62a19a1c58","sha256:f7144dc41f6ff916dbcb0ea46abce2ff1a9e04f4048291501fc2257a6cf8d85c"],"state_sha256":"b5a26ea874fd3289ecb68ad434ed58bff515fd1dbdf639c9a07dee9c1916cdf3"}