{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:YV3TZM2NJDOEBOGMK37K74G52Q","short_pith_number":"pith:YV3TZM2N","schema_version":"1.0","canonical_sha256":"c5773cb34d48dc40b8cc56feaff0ddd421e6cce86215f8c16feb341476f65da0","source":{"kind":"arxiv","id":"2506.11210","version":1},"attestation_state":"computed","paper":{"title":"Second-Order Parameterizations for the Complexity Theory of Integrable Functions","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Aras Bacho, Martin Ziegler","submitted_at":"2025-06-12T18:18:48Z","abstract_excerpt":"We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of second-order parameterized complexity theory for spaces of continuous functions. Specifically we prove the mutual linear equivalence of three natural parameterizations of the space $\\Lrm{p}$ of $p$-integrable complex functions on the real unit interval: (binary) $\\Lrm{p}$-modulus, rate of convergence of Fourier series, and rate of approximation by step functions."},"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":"2506.11210","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.CC","submitted_at":"2025-06-12T18:18:48Z","cross_cats_sorted":[],"title_canon_sha256":"c6c37acd7704dd2413918f2929aa9196f3ef689a46b2e5ae12d0f829aaff960a","abstract_canon_sha256":"93b28dc7b16f8cb5530b3fe8b1363df1c7074dd7521761f8c46c847dbb6335cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:01.234250Z","signature_b64":"rKg0fce+eAEpKSNrg1K1ZDAOxQVw83oJXec/bNdF22HKbk7vjPgG4G60N4/eFIV49cnoTZbfV9QtbgNuvsUKAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5773cb34d48dc40b8cc56feaff0ddd421e6cce86215f8c16feb341476f65da0","last_reissued_at":"2026-07-05T11:21:01.233847Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:01.233847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Second-Order Parameterizations for the Complexity Theory of Integrable Functions","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Aras Bacho, Martin Ziegler","submitted_at":"2025-06-12T18:18:48Z","abstract_excerpt":"We develop a unified second-order parameterized complexity theory for spaces of integrable functions. This generalizes the well-established case of second-order parameterized complexity theory for spaces of continuous functions. Specifically we prove the mutual linear equivalence of three natural parameterizations of the space $\\Lrm{p}$ of $p$-integrable complex functions on the real unit interval: (binary) $\\Lrm{p}$-modulus, rate of convergence of Fourier series, and rate of approximation by step functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.11210","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.11210/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.11210","created_at":"2026-07-05T11:21:01.233903+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.11210v1","created_at":"2026-07-05T11:21:01.233903+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.11210","created_at":"2026-07-05T11:21:01.233903+00:00"},{"alias_kind":"pith_short_12","alias_value":"YV3TZM2NJDOE","created_at":"2026-07-05T11:21:01.233903+00:00"},{"alias_kind":"pith_short_16","alias_value":"YV3TZM2NJDOEBOGM","created_at":"2026-07-05T11:21:01.233903+00:00"},{"alias_kind":"pith_short_8","alias_value":"YV3TZM2N","created_at":"2026-07-05T11:21:01.233903+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/YV3TZM2NJDOEBOGMK37K74G52Q","json":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q.json","graph_json":"https://pith.science/api/pith-number/YV3TZM2NJDOEBOGMK37K74G52Q/graph.json","events_json":"https://pith.science/api/pith-number/YV3TZM2NJDOEBOGMK37K74G52Q/events.json","paper":"https://pith.science/paper/YV3TZM2N"},"agent_actions":{"view_html":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q","download_json":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q.json","view_paper":"https://pith.science/paper/YV3TZM2N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.11210&json=true","fetch_graph":"https://pith.science/api/pith-number/YV3TZM2NJDOEBOGMK37K74G52Q/graph.json","fetch_events":"https://pith.science/api/pith-number/YV3TZM2NJDOEBOGMK37K74G52Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q/action/storage_attestation","attest_author":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q/action/author_attestation","sign_citation":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q/action/citation_signature","submit_replication":"https://pith.science/pith/YV3TZM2NJDOEBOGMK37K74G52Q/action/replication_record"}},"created_at":"2026-07-05T11:21:01.233903+00:00","updated_at":"2026-07-05T11:21:01.233903+00:00"}