{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:VA5UAONC3MX6COMC5TON2XEEPX","short_pith_number":"pith:VA5UAONC","schema_version":"1.0","canonical_sha256":"a83b4039a2db2fe13982ecdcdd5c847df48167af893717fe28ed22b0446cf21d","source":{"kind":"arxiv","id":"1603.02064","version":1},"attestation_state":"computed","paper":{"title":"On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","submitted_at":"2016-02-22T11:16:44Z","abstract_excerpt":"Let $G$ be a locally compact group, and let $K$ be a compact subgroup of $G$. Let $\\mu : G\\longrightarrow\\mathbb{C}\\backslash\\{0\\}$ be a character of $G$. In this paper, we deal with the integral equations $$W_{\\mu}(K):\\; \\;\\int_{K}f(xkyk^{-1})dk+\\mu(y)\\int_{K}f(xky^{-1}k^{-1})dk=2f(x)g(y),$$ and $$D_{\\mu}(K):\\; \\;\\int_{K}f(xkyk^{-1})dk+\\mu(y)\\int_{K}f(xky^{-1}k^{-1})dk=2f(x)f(y)$$ for all $x, y\\in G$ where $f, g: G\\longrightarrow \\mathbb{C}$, to be determined, are complex continuous functions on $G$.\n  When $K\\subset Z(G)$, the center of $G$, $D_{\\mu}(K)$ reduces to the new version of d'Almbe"},"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":"1603.02064","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-02-22T11:16:44Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"93b846483d254d00955d3b1a50538f2cfed52472990aef8f3e467973227129a8","abstract_canon_sha256":"cd23c427dfe89e7411f2db82b4781418fa6e196d73e3ded0199235aaaed1d8bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:31.725528Z","signature_b64":"H7AH6QBfQ1uTE1i6WPbLiI8CaWv91a3wGKKzWCXo2D8wXXbsUcgyRIpOgleAm+jC1uzXBGWTmBMFkz7dAjXkCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a83b4039a2db2fe13982ecdcdd5c847df48167af893717fe28ed22b0446cf21d","last_reissued_at":"2026-05-18T01:19:31.724947Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:31.724947Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the integral functional equations: On the integral d'Alembert's and Wilson's functional equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","submitted_at":"2016-02-22T11:16:44Z","abstract_excerpt":"Let $G$ be a locally compact group, and let $K$ be a compact subgroup of $G$. Let $\\mu : G\\longrightarrow\\mathbb{C}\\backslash\\{0\\}$ be a character of $G$. In this paper, we deal with the integral equations $$W_{\\mu}(K):\\; \\;\\int_{K}f(xkyk^{-1})dk+\\mu(y)\\int_{K}f(xky^{-1}k^{-1})dk=2f(x)g(y),$$ and $$D_{\\mu}(K):\\; \\;\\int_{K}f(xkyk^{-1})dk+\\mu(y)\\int_{K}f(xky^{-1}k^{-1})dk=2f(x)f(y)$$ for all $x, y\\in G$ where $f, g: G\\longrightarrow \\mathbb{C}$, to be determined, are complex continuous functions on $G$.\n  When $K\\subset Z(G)$, the center of $G$, $D_{\\mu}(K)$ reduces to the new version of d'Almbe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.02064","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":"1603.02064","created_at":"2026-05-18T01:19:31.725039+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.02064v1","created_at":"2026-05-18T01:19:31.725039+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.02064","created_at":"2026-05-18T01:19:31.725039+00:00"},{"alias_kind":"pith_short_12","alias_value":"VA5UAONC3MX6","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"VA5UAONC3MX6COMC","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"VA5UAONC","created_at":"2026-05-18T12:30:48.956258+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/VA5UAONC3MX6COMC5TON2XEEPX","json":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX.json","graph_json":"https://pith.science/api/pith-number/VA5UAONC3MX6COMC5TON2XEEPX/graph.json","events_json":"https://pith.science/api/pith-number/VA5UAONC3MX6COMC5TON2XEEPX/events.json","paper":"https://pith.science/paper/VA5UAONC"},"agent_actions":{"view_html":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX","download_json":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX.json","view_paper":"https://pith.science/paper/VA5UAONC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.02064&json=true","fetch_graph":"https://pith.science/api/pith-number/VA5UAONC3MX6COMC5TON2XEEPX/graph.json","fetch_events":"https://pith.science/api/pith-number/VA5UAONC3MX6COMC5TON2XEEPX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX/action/storage_attestation","attest_author":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX/action/author_attestation","sign_citation":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX/action/citation_signature","submit_replication":"https://pith.science/pith/VA5UAONC3MX6COMC5TON2XEEPX/action/replication_record"}},"created_at":"2026-05-18T01:19:31.725039+00:00","updated_at":"2026-05-18T01:19:31.725039+00:00"}