{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:DSU3AUVHZUW6ROU5JP5MDR2TRZ","short_pith_number":"pith:DSU3AUVH","schema_version":"1.0","canonical_sha256":"1ca9b052a7cd2de8ba9d4bfac1c7538e7bd9507031004cdf22b3f37438fcd5c9","source":{"kind":"arxiv","id":"1103.1950","version":3},"attestation_state":"computed","paper":{"title":"The Lebesgue Constant for the Periodic Franklin System","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Markus Passenbrunner","submitted_at":"2011-03-10T07:02:03Z","abstract_excerpt":"We identify the torus with the unit interval $[0,1)$ and let $n,\\nu\\in\\mathbb{N}$, $1\\leq \\nu\\leq n-1$ and $N:=n+\\nu$. Then we define the (partially equally spaced) knots \\[ t_{j}=\\{[c]{ll}% \\frac{j}{2n}, & \\text{for}j=0,...,2\\nu, \\frac{j-\\nu}{n}, & \\text{for}j=2\\nu+1,...,N-1.] Furthermore, given $n,\\nu$ we let $V_{n,\\nu}$ be the space of piecewise linear continuous functions on the torus with knots $\\{t_j:0\\leq j\\leq N-1\\}$. Finally, let $P_{n,\\nu}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,\\nu}.$ The main result is \\[\\lim_{n\\rightarrow\\infty,\\nu=1}\\|P_{n,\\nu}:L^\\inft"},"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":"1103.1950","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-03-10T07:02:03Z","cross_cats_sorted":[],"title_canon_sha256":"bcb657b4bd67a0da8d992a0db588271659139c57b7ef6ee459f024c8521cf23e","abstract_canon_sha256":"b035f6874ec5075d4089340a87222707008f64116b01e6c42f8e4f75c007487d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:49.405551Z","signature_b64":"+ax4CBBCwzgZhWTaHg52JMcs22+6PiN54HNGL2tji2hrndem4iR+v1Y0TSTeXARY+evcLGDURyUNlOPOsBKiDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ca9b052a7cd2de8ba9d4bfac1c7538e7bd9507031004cdf22b3f37438fcd5c9","last_reissued_at":"2026-05-18T02:25:49.405129Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:49.405129Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Lebesgue Constant for the Periodic Franklin System","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Markus Passenbrunner","submitted_at":"2011-03-10T07:02:03Z","abstract_excerpt":"We identify the torus with the unit interval $[0,1)$ and let $n,\\nu\\in\\mathbb{N}$, $1\\leq \\nu\\leq n-1$ and $N:=n+\\nu$. Then we define the (partially equally spaced) knots \\[ t_{j}=\\{[c]{ll}% \\frac{j}{2n}, & \\text{for}j=0,...,2\\nu, \\frac{j-\\nu}{n}, & \\text{for}j=2\\nu+1,...,N-1.] Furthermore, given $n,\\nu$ we let $V_{n,\\nu}$ be the space of piecewise linear continuous functions on the torus with knots $\\{t_j:0\\leq j\\leq N-1\\}$. Finally, let $P_{n,\\nu}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,\\nu}.$ The main result is \\[\\lim_{n\\rightarrow\\infty,\\nu=1}\\|P_{n,\\nu}:L^\\inft"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1950","kind":"arxiv","version":3},"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":"1103.1950","created_at":"2026-05-18T02:25:49.405185+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.1950v3","created_at":"2026-05-18T02:25:49.405185+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.1950","created_at":"2026-05-18T02:25:49.405185+00:00"},{"alias_kind":"pith_short_12","alias_value":"DSU3AUVHZUW6","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_16","alias_value":"DSU3AUVHZUW6ROU5","created_at":"2026-05-18T12:26:26.731475+00:00"},{"alias_kind":"pith_short_8","alias_value":"DSU3AUVH","created_at":"2026-05-18T12:26:26.731475+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/DSU3AUVHZUW6ROU5JP5MDR2TRZ","json":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ.json","graph_json":"https://pith.science/api/pith-number/DSU3AUVHZUW6ROU5JP5MDR2TRZ/graph.json","events_json":"https://pith.science/api/pith-number/DSU3AUVHZUW6ROU5JP5MDR2TRZ/events.json","paper":"https://pith.science/paper/DSU3AUVH"},"agent_actions":{"view_html":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ","download_json":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ.json","view_paper":"https://pith.science/paper/DSU3AUVH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.1950&json=true","fetch_graph":"https://pith.science/api/pith-number/DSU3AUVHZUW6ROU5JP5MDR2TRZ/graph.json","fetch_events":"https://pith.science/api/pith-number/DSU3AUVHZUW6ROU5JP5MDR2TRZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ/action/storage_attestation","attest_author":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ/action/author_attestation","sign_citation":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ/action/citation_signature","submit_replication":"https://pith.science/pith/DSU3AUVHZUW6ROU5JP5MDR2TRZ/action/replication_record"}},"created_at":"2026-05-18T02:25:49.405185+00:00","updated_at":"2026-05-18T02:25:49.405185+00:00"}