{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:2NBSK6L6IJ2ZKV7CUQKBI2XPGI","short_pith_number":"pith:2NBSK6L6","canonical_record":{"source":{"id":"1310.2283","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-10-08T21:32:18Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"ceb72c9222394aad2b0aea27488c15d9ba394da610d5c0638c7b987278123e33","abstract_canon_sha256":"13a3b73d08bc6d750b3e6894ba238f5144df11d4b1f9879a0093c21904d05b55"},"schema_version":"1.0"},"canonical_sha256":"d34325797e42759557e2a414146aef320db4ffe5a06a86d68fe883bafea1a404","source":{"kind":"arxiv","id":"1310.2283","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.2283","created_at":"2026-05-18T03:07:42Z"},{"alias_kind":"arxiv_version","alias_value":"1310.2283v2","created_at":"2026-05-18T03:07:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2283","created_at":"2026-05-18T03:07:42Z"},{"alias_kind":"pith_short_12","alias_value":"2NBSK6L6IJ2Z","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2NBSK6L6IJ2ZKV7C","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2NBSK6L6","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:2NBSK6L6IJ2ZKV7CUQKBI2XPGI","target":"record","payload":{"canonical_record":{"source":{"id":"1310.2283","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-10-08T21:32:18Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"ceb72c9222394aad2b0aea27488c15d9ba394da610d5c0638c7b987278123e33","abstract_canon_sha256":"13a3b73d08bc6d750b3e6894ba238f5144df11d4b1f9879a0093c21904d05b55"},"schema_version":"1.0"},"canonical_sha256":"d34325797e42759557e2a414146aef320db4ffe5a06a86d68fe883bafea1a404","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:42.106872Z","signature_b64":"7JQ9owZ3Sz6l25/lMyhE9IWZwTOHAF4uM46QzxOnl+cBfZ3MCA/GRNZjoc+anhg3dYaeQsfn4Z189WAw3mjIBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d34325797e42759557e2a414146aef320db4ffe5a06a86d68fe883bafea1a404","last_reissued_at":"2026-05-18T03:07:42.106191Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:42.106191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.2283","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+SNwsoTni907eSGq9IGS79wM+h2For2tnw9/eJ58qlYjCxXryWxtEKkJAd/7i43WiWQQ++1ttuCFQDfHFJtsCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T10:11:15.397589Z"},"content_sha256":"ac055b7e3f002d1cfa1e72cb76bf64fad541e841adf6e209c20e37efe06ba3d1","schema_version":"1.0","event_id":"sha256:ac055b7e3f002d1cfa1e72cb76bf64fad541e841adf6e209c20e37efe06ba3d1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:2NBSK6L6IJ2ZKV7CUQKBI2XPGI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral approximation on the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.CA","authors_text":"Huiyuan Li, Yuan Xu","submitted_at":"2013-10-08T21:32:18Z","abstract_excerpt":"Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\\ball)$, $1<p<\\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces and explicit construction of approximating polynomials. One major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the Sobolev space $W_2^s(\\ball)$. As an application, a direct and efficient spectral-Galerkin method based on our orthogonal polynomials is proposed for the second and the fourth order ellipt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2283","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:07:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lTrLP132y8IxLFvKPpN6LmuaGbM4rdos0ZrAVwWaw3L9+1T4TtVqqfB3+QZEJKO3ClOLwtMr3WZC26wUu7/bCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T10:11:15.397942Z"},"content_sha256":"937d78641a3cf44280963d92c4572b47eb3f43eb472fcc06a3b6b01d79c2ddc1","schema_version":"1.0","event_id":"sha256:937d78641a3cf44280963d92c4572b47eb3f43eb472fcc06a3b6b01d79c2ddc1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI/bundle.json","state_url":"https://pith.science/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI/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":"2026-06-02T10:11:15Z","links":{"resolver":"https://pith.science/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI","bundle":"https://pith.science/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI/bundle.json","state":"https://pith.science/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2NBSK6L6IJ2ZKV7CUQKBI2XPGI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2NBSK6L6IJ2ZKV7CUQKBI2XPGI","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":"13a3b73d08bc6d750b3e6894ba238f5144df11d4b1f9879a0093c21904d05b55","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-10-08T21:32:18Z","title_canon_sha256":"ceb72c9222394aad2b0aea27488c15d9ba394da610d5c0638c7b987278123e33"},"schema_version":"1.0","source":{"id":"1310.2283","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.2283","created_at":"2026-05-18T03:07:42Z"},{"alias_kind":"arxiv_version","alias_value":"1310.2283v2","created_at":"2026-05-18T03:07:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2283","created_at":"2026-05-18T03:07:42Z"},{"alias_kind":"pith_short_12","alias_value":"2NBSK6L6IJ2Z","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2NBSK6L6IJ2ZKV7C","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2NBSK6L6","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:937d78641a3cf44280963d92c4572b47eb3f43eb472fcc06a3b6b01d79c2ddc1","target":"graph","created_at":"2026-05-18T03:07:42Z","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":"Spectral approximation by polynomials on the unit ball is studied in the frame of the Sobolev spaces $W^{s}_p(\\ball)$, $1<p<\\infty$. The main results give sharp estimates on the order of approximation by polynomials in the Sobolev spaces and explicit construction of approximating polynomials. One major effort lies in understanding the structure of orthogonal polynomials with respect to an inner product of the Sobolev space $W_2^s(\\ball)$. As an application, a direct and efficient spectral-Galerkin method based on our orthogonal polynomials is proposed for the second and the fourth order ellipt","authors_text":"Huiyuan Li, Yuan Xu","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-10-08T21:32:18Z","title":"Spectral approximation on the unit ball"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2283","kind":"arxiv","version":2},"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:ac055b7e3f002d1cfa1e72cb76bf64fad541e841adf6e209c20e37efe06ba3d1","target":"record","created_at":"2026-05-18T03:07:42Z","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":"13a3b73d08bc6d750b3e6894ba238f5144df11d4b1f9879a0093c21904d05b55","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-10-08T21:32:18Z","title_canon_sha256":"ceb72c9222394aad2b0aea27488c15d9ba394da610d5c0638c7b987278123e33"},"schema_version":"1.0","source":{"id":"1310.2283","kind":"arxiv","version":2}},"canonical_sha256":"d34325797e42759557e2a414146aef320db4ffe5a06a86d68fe883bafea1a404","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d34325797e42759557e2a414146aef320db4ffe5a06a86d68fe883bafea1a404","first_computed_at":"2026-05-18T03:07:42.106191Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:42.106191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7JQ9owZ3Sz6l25/lMyhE9IWZwTOHAF4uM46QzxOnl+cBfZ3MCA/GRNZjoc+anhg3dYaeQsfn4Z189WAw3mjIBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:42.106872Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.2283","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac055b7e3f002d1cfa1e72cb76bf64fad541e841adf6e209c20e37efe06ba3d1","sha256:937d78641a3cf44280963d92c4572b47eb3f43eb472fcc06a3b6b01d79c2ddc1"],"state_sha256":"7eecf73b2780e3a2c477bcf69d3989f1b90dc2aaa3225647f375f88a2684f310"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l/RDlPabRocvae/NuGWkZAxbNsZUNcQ/akaygDmLAQDeTfco1EWb8BkfXPnEBvBkW+erNxHOdgJmG60DdJEvCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T10:11:15.400024Z","bundle_sha256":"ff204269335ceb12388cb095cea1206dbf8e1db9b70619f8dd1fa9b89b95f306"}}