{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:UKTGWKJMC75AVFNU75BALNQSI6","short_pith_number":"pith:UKTGWKJM","schema_version":"1.0","canonical_sha256":"a2a66b292c17fa0a95b4ff4205b61247a4ffb9e897c7a13d6bae31b8047decd1","source":{"kind":"arxiv","id":"1103.4380","version":2},"attestation_state":"computed","paper":{"title":"Divergence of mock and scrambled Fourier series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Deguang Han, Dorin Ervin Dutkay, Qiyu Sun","submitted_at":"2011-03-22T20:52:09Z","abstract_excerpt":"We study divergence properties of Fourier series on Cantor-type fractal measures, also called mock Fourier series. We show that in some cases the $L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, which we call scrambled Fourier series, have a corresponding Dirichlet kernel whose $L^1$-norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence propert"},"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":"1103.4380","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-22T20:52:09Z","cross_cats_sorted":[],"title_canon_sha256":"58062f3ae4d7f08d970e3b3fbfb2939c729958761f0c17a76d06b0661f0e8aa2","abstract_canon_sha256":"ccbb71ee4aade28803cea85bfb8613e7113b7a93c03e41975b1d0e446232e9e1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:18.681342Z","signature_b64":"qYnHVYIP3zBbAWDWPGy2IPkc08myhtx+Thoj3fhSexrQSJl4TNUCvco0lDMVHVQkt4WFKv1IAQvKRdUtU2KfDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2a66b292c17fa0a95b4ff4205b61247a4ffb9e897c7a13d6bae31b8047decd1","last_reissued_at":"2026-05-18T03:46:18.680753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:18.680753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Divergence of mock and scrambled Fourier series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Deguang Han, Dorin Ervin Dutkay, Qiyu Sun","submitted_at":"2011-03-22T20:52:09Z","abstract_excerpt":"We study divergence properties of Fourier series on Cantor-type fractal measures, also called mock Fourier series. We show that in some cases the $L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, which we call scrambled Fourier series, have a corresponding Dirichlet kernel whose $L^1$-norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence propert"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.4380","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":"1103.4380","created_at":"2026-05-18T03:46:18.680842+00:00"},{"alias_kind":"arxiv_version","alias_value":"1103.4380v2","created_at":"2026-05-18T03:46:18.680842+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.4380","created_at":"2026-05-18T03:46:18.680842+00:00"},{"alias_kind":"pith_short_12","alias_value":"UKTGWKJMC75A","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_16","alias_value":"UKTGWKJMC75AVFNU","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_8","alias_value":"UKTGWKJM","created_at":"2026-05-18T12:26:42.757692+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/UKTGWKJMC75AVFNU75BALNQSI6","json":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6.json","graph_json":"https://pith.science/api/pith-number/UKTGWKJMC75AVFNU75BALNQSI6/graph.json","events_json":"https://pith.science/api/pith-number/UKTGWKJMC75AVFNU75BALNQSI6/events.json","paper":"https://pith.science/paper/UKTGWKJM"},"agent_actions":{"view_html":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6","download_json":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6.json","view_paper":"https://pith.science/paper/UKTGWKJM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1103.4380&json=true","fetch_graph":"https://pith.science/api/pith-number/UKTGWKJMC75AVFNU75BALNQSI6/graph.json","fetch_events":"https://pith.science/api/pith-number/UKTGWKJMC75AVFNU75BALNQSI6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6/action/storage_attestation","attest_author":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6/action/author_attestation","sign_citation":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6/action/citation_signature","submit_replication":"https://pith.science/pith/UKTGWKJMC75AVFNU75BALNQSI6/action/replication_record"}},"created_at":"2026-05-18T03:46:18.680842+00:00","updated_at":"2026-05-18T03:46:18.680842+00:00"}