{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:3SALY5HSQPSM2QAG62JRLREKZI","short_pith_number":"pith:3SALY5HS","schema_version":"1.0","canonical_sha256":"dc80bc74f283e4cd4006f69315c48aca17eadf9e71ce09be4202292ee281f55b","source":{"kind":"arxiv","id":"1808.04313","version":1},"attestation_state":"computed","paper":{"title":"Fourier transform inversion using an elementary differential equation and a contour integral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Erik Talvila","submitted_at":"2018-08-13T16:15:09Z","abstract_excerpt":"Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by $f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires elementary manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little ma"},"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":"1808.04313","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-08-13T16:15:09Z","cross_cats_sorted":[],"title_canon_sha256":"a105aa356cca31a17db934bf8f0907aaf08aeef41ca64a1af458203844c15ee8","abstract_canon_sha256":"b1020f6b4755244281d61c1ca4852373650ed282508b52c9ef1786a60733a44c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:16.721536Z","signature_b64":"xGdnZe48D7zuqB/Y4jMP9gmF8R3AJEEHLO3RHdowRsW8FL/Mq1PjC4u+3e/nc5rHiec3XOuAF/qrubFsNgLWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc80bc74f283e4cd4006f69315c48aca17eadf9e71ce09be4202292ee281f55b","last_reissued_at":"2026-05-18T00:08:16.720855Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:16.720855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fourier transform inversion using an elementary differential equation and a contour integral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Erik Talvila","submitted_at":"2018-08-13T16:15:09Z","abstract_excerpt":"Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by $f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires elementary manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.04313","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":""},"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":"1808.04313","created_at":"2026-05-18T00:08:16.720951+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.04313v1","created_at":"2026-05-18T00:08:16.720951+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.04313","created_at":"2026-05-18T00:08:16.720951+00:00"},{"alias_kind":"pith_short_12","alias_value":"3SALY5HSQPSM","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"3SALY5HSQPSM2QAG","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"3SALY5HS","created_at":"2026-05-18T12:32:05.422762+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/3SALY5HSQPSM2QAG62JRLREKZI","json":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI.json","graph_json":"https://pith.science/api/pith-number/3SALY5HSQPSM2QAG62JRLREKZI/graph.json","events_json":"https://pith.science/api/pith-number/3SALY5HSQPSM2QAG62JRLREKZI/events.json","paper":"https://pith.science/paper/3SALY5HS"},"agent_actions":{"view_html":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI","download_json":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI.json","view_paper":"https://pith.science/paper/3SALY5HS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.04313&json=true","fetch_graph":"https://pith.science/api/pith-number/3SALY5HSQPSM2QAG62JRLREKZI/graph.json","fetch_events":"https://pith.science/api/pith-number/3SALY5HSQPSM2QAG62JRLREKZI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI/action/storage_attestation","attest_author":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI/action/author_attestation","sign_citation":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI/action/citation_signature","submit_replication":"https://pith.science/pith/3SALY5HSQPSM2QAG62JRLREKZI/action/replication_record"}},"created_at":"2026-05-18T00:08:16.720951+00:00","updated_at":"2026-05-18T00:08:16.720951+00:00"}