{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:HVK6BPF6VXZW4BKKYHT3NFPUDI","short_pith_number":"pith:HVK6BPF6","canonical_record":{"source":{"id":"1806.08105","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-21T08:28:33Z","cross_cats_sorted":[],"title_canon_sha256":"fc5b169e0da3ba885d581b6cb7a6610f490cb69cd68475a5753e1aa46d72cbe3","abstract_canon_sha256":"fe23cbe2921308e86803fef90dff98b3810d8b2f2b9276b308982e31605e3303"},"schema_version":"1.0"},"canonical_sha256":"3d55e0bcbeadf36e054ac1e7b695f41a181db83536cdbb8c1b6104ab14f8a723","source":{"kind":"arxiv","id":"1806.08105","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.08105","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"arxiv_version","alias_value":"1806.08105v2","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.08105","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"pith_short_12","alias_value":"HVK6BPF6VXZW","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HVK6BPF6VXZW4BKK","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HVK6BPF6","created_at":"2026-05-18T12:32:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:HVK6BPF6VXZW4BKKYHT3NFPUDI","target":"record","payload":{"canonical_record":{"source":{"id":"1806.08105","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-21T08:28:33Z","cross_cats_sorted":[],"title_canon_sha256":"fc5b169e0da3ba885d581b6cb7a6610f490cb69cd68475a5753e1aa46d72cbe3","abstract_canon_sha256":"fe23cbe2921308e86803fef90dff98b3810d8b2f2b9276b308982e31605e3303"},"schema_version":"1.0"},"canonical_sha256":"3d55e0bcbeadf36e054ac1e7b695f41a181db83536cdbb8c1b6104ab14f8a723","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:37.641584Z","signature_b64":"c5sv0Izr5/ZC/0x6LTW04TzSaWLSXpf4SK5DMILnOLy6Sk2wQHTkLbKnxudhefxSxtfVNgIgZOELlr9C9P6KDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d55e0bcbeadf36e054ac1e7b695f41a181db83536cdbb8c1b6104ab14f8a723","last_reissued_at":"2026-05-18T00:12:37.640957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:37.640957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1806.08105","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":"2026-05-18T00:12:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oJcKm8zm6OR1K2TD/x2GaGE878iGOR379cAhowl/P36o5rgYeemsW0cY0RMddyqQX7u8xFXOEisCfkZsN3reBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T13:25:10.030049Z"},"content_sha256":"7309f6ebd191cc68cef63cd687e51f3d687192490a37cd112faa94bcecdc494e","schema_version":"1.0","event_id":"sha256:7309f6ebd191cc68cef63cd687e51f3d687192490a37cd112faa94bcecdc494e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:HVK6BPF6VXZW4BKKYHT3NFPUDI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fourier series of Jacobi-Sobolev polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Judit M\\'inguez, \\'Oscar Ciaurri","submitted_at":"2018-06-21T08:28:33Z","abstract_excerpt":"Let $\\{q_n^{(\\alpha,\\beta,m)}(x)\\}_{n\\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \\begin{equation*} \\langle f,g\\rangle_{\\alpha,\\beta,m}=\\sum_{k=0}^m \\int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\\, dw_{\\alpha+k,\\beta+k}(x), \\quad \\alpha,\\beta>-1, \\quad m\\ge 1, \\end{equation*} where $dw_{a,b}(x)=(1-x)^{a}(1+x)^b\\, dx$. We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space $W_{\\alpha,\\beta}^{p,m}$. As a consequence we deduce the convergence of such partial sums in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08105","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":"2026-05-18T00:12:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jZ1zrY4L33NV80qCUscT4YaP0citOSLCb/1/RsxB/sjts7ZZKY0eJD/76ddxbpY9CdHwQPgdsvpLmgl260b/CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T13:25:10.030420Z"},"content_sha256":"1fb91ed14afdc50252b76c31ccbc1b7db723e2bf76ccecdb4c4f592dbf3519d1","schema_version":"1.0","event_id":"sha256:1fb91ed14afdc50252b76c31ccbc1b7db723e2bf76ccecdb4c4f592dbf3519d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/bundle.json","state_url":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/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":"2026-06-24T13:25:10Z","links":{"resolver":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI","bundle":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/bundle.json","state":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:HVK6BPF6VXZW4BKKYHT3NFPUDI","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":"fe23cbe2921308e86803fef90dff98b3810d8b2f2b9276b308982e31605e3303","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-21T08:28:33Z","title_canon_sha256":"fc5b169e0da3ba885d581b6cb7a6610f490cb69cd68475a5753e1aa46d72cbe3"},"schema_version":"1.0","source":{"id":"1806.08105","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.08105","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"arxiv_version","alias_value":"1806.08105v2","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.08105","created_at":"2026-05-18T00:12:37Z"},{"alias_kind":"pith_short_12","alias_value":"HVK6BPF6VXZW","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_16","alias_value":"HVK6BPF6VXZW4BKK","created_at":"2026-05-18T12:32:28Z"},{"alias_kind":"pith_short_8","alias_value":"HVK6BPF6","created_at":"2026-05-18T12:32:28Z"}],"graph_snapshots":[{"event_id":"sha256:1fb91ed14afdc50252b76c31ccbc1b7db723e2bf76ccecdb4c4f592dbf3519d1","target":"graph","created_at":"2026-05-18T00:12:37Z","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":"Let $\\{q_n^{(\\alpha,\\beta,m)}(x)\\}_{n\\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \\begin{equation*} \\langle f,g\\rangle_{\\alpha,\\beta,m}=\\sum_{k=0}^m \\int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\\, dw_{\\alpha+k,\\beta+k}(x), \\quad \\alpha,\\beta>-1, \\quad m\\ge 1, \\end{equation*} where $dw_{a,b}(x)=(1-x)^{a}(1+x)^b\\, dx$. We obtain necessary and sufficient conditions for the uniform boundedness of the partial sum operators related to this sequence of polynomials in the Sobolev space $W_{\\alpha,\\beta}^{p,m}$. As a consequence we deduce the convergence of such partial sums in ","authors_text":"Judit M\\'inguez, \\'Oscar Ciaurri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-21T08:28:33Z","title":"Fourier series of Jacobi-Sobolev polynomial"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08105","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":"sha256:7309f6ebd191cc68cef63cd687e51f3d687192490a37cd112faa94bcecdc494e","target":"record","created_at":"2026-05-18T00:12:37Z","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":"fe23cbe2921308e86803fef90dff98b3810d8b2f2b9276b308982e31605e3303","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-06-21T08:28:33Z","title_canon_sha256":"fc5b169e0da3ba885d581b6cb7a6610f490cb69cd68475a5753e1aa46d72cbe3"},"schema_version":"1.0","source":{"id":"1806.08105","kind":"arxiv","version":2}},"canonical_sha256":"3d55e0bcbeadf36e054ac1e7b695f41a181db83536cdbb8c1b6104ab14f8a723","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3d55e0bcbeadf36e054ac1e7b695f41a181db83536cdbb8c1b6104ab14f8a723","first_computed_at":"2026-05-18T00:12:37.640957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:37.640957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c5sv0Izr5/ZC/0x6LTW04TzSaWLSXpf4SK5DMILnOLy6Sk2wQHTkLbKnxudhefxSxtfVNgIgZOELlr9C9P6KDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:37.641584Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.08105","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7309f6ebd191cc68cef63cd687e51f3d687192490a37cd112faa94bcecdc494e","sha256:1fb91ed14afdc50252b76c31ccbc1b7db723e2bf76ccecdb4c4f592dbf3519d1"],"state_sha256":"292bee840cad9625d333e5fc360957cf0aabf6a50a015911151332b20be9a96e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TMP0H/S2vF0eC9Siy/r8Zv+zsI6yaSS0SxMm9rlA142fxa1bvdbxSOKRkIM1W98tDWpuq9AdF7jBOGe6KJFZAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T13:25:10.032391Z","bundle_sha256":"e3e7a93e0e76384e27e35e254ceb148119c26f40f01804bd6e58274e034a33b9"}}