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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 "},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1806.08105","created_at":"2026-05-18T00:12:37.641069+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.08105v2","created_at":"2026-05-18T00:12:37.641069+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.08105","created_at":"2026-05-18T00:12:37.641069+00:00"},{"alias_kind":"pith_short_12","alias_value":"HVK6BPF6VXZW","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"HVK6BPF6VXZW4BKK","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"HVK6BPF6","created_at":"2026-05-18T12:32:28.185984+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/HVK6BPF6VXZW4BKKYHT3NFPUDI","json":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI.json","graph_json":"https://pith.science/api/pith-number/HVK6BPF6VXZW4BKKYHT3NFPUDI/graph.json","events_json":"https://pith.science/api/pith-number/HVK6BPF6VXZW4BKKYHT3NFPUDI/events.json","paper":"https://pith.science/paper/HVK6BPF6"},"agent_actions":{"view_html":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI","download_json":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI.json","view_paper":"https://pith.science/paper/HVK6BPF6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.08105&json=true","fetch_graph":"https://pith.science/api/pith-number/HVK6BPF6VXZW4BKKYHT3NFPUDI/graph.json","fetch_events":"https://pith.science/api/pith-number/HVK6BPF6VXZW4BKKYHT3NFPUDI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/action/storage_attestation","attest_author":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/action/author_attestation","sign_citation":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/action/citation_signature","submit_replication":"https://pith.science/pith/HVK6BPF6VXZW4BKKYHT3NFPUDI/action/replication_record"}},"created_at":"2026-05-18T00:12:37.641069+00:00","updated_at":"2026-05-18T00:12:37.641069+00:00"}