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We show that the solutions of these equations are closely related to the solutions of the d'Alembert's classic functional equ"},"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":"1607.05166","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-07-12T10:49:44Z","cross_cats_sorted":[],"title_canon_sha256":"5e892469e45412c57fc805ce646c054bcb75db33d908b9256a8ef5cfb785ee1f","abstract_canon_sha256":"b993536e23eb7224154c963f550598a46dd6786fd10d50dce2817dfd1eeb80f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:55.748139Z","signature_b64":"CsPtyJMA+fgHhIUIBylzqnSMe6VtrF7AbzywkKbQWX8HvhTmWwfiVvPNDyZEXO9qj0jbYIKaDeyBs/gNzsKSCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"140531b47d1ce14bec7e86f99273cf254c6998cce4d757775704d35482dce308","last_reissued_at":"2026-05-18T01:10:55.747550Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:55.747550Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Solutions and stability of generalized Kannappan's and Van Vleck's functional equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Elqorachi Elhoucien, Redouani Ahmed","submitted_at":"2016-07-12T10:49:44Z","abstract_excerpt":"We study the solutions of the integral Kannappan's and Van Vleck's functional equations $$\\int_{S}f(xyt)d\\mu(t)+\\int_{S}f(x\\sigma(y)t)d\\mu(t) = 2f(x)f(y), \\;x,y\\in S;$$ $$\\int_{S}f( x\\sigma(y)t)d\\mu(t)-\\int_{S}f(xyt)d\\mu(t) = 2f(x)f(y), \\;x,y\\in S,$$ where $S$ is a semigroup, $\\sigma$ is an involutive automorphism of $S$ and $\\mu$ is a linear combination of Dirac measures $(\\delta_{z_{i}})_{i\\in I}$, such that for all $i\\in I$, $z_{i}$ is contained in the center of $S$. 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