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Molchanov","submitted_at":"2014-09-22T09:20:45Z","abstract_excerpt":"We study the archetypal functional equation of the form $y(x)=\\iint_{\\mathbb{R}^2} y(a(x-b))\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)$ ($x\\in\\mathbb{R}$), where $\\mu$ is a probability measure on $\\mathbb{R}^2$; equivalently, $y(x)=\\mathbb{E}\\{y(\\alpha(x-\\beta))\\}$, where $\\mathbb{E}$ is expectation with respect to the distribution $\\mu$ of random coefficients $(\\alpha,\\beta)$. Existence of non-trivial (i.e., non-constant) bounded continuous solutions is governed by the value $K:=\\iint_{\\mathbb{R}^2}\\ln|a|\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)=\\mathbb{E}\\{\\ln|\\alpha|\\}$; namely, under mild technical conditions no"},"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":"1409.6126","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-09-22T09:20:45Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"c1e9928f7fcaf304a91ebba4c3bc6d4323c59844a4b9247e28c9272a751ed55c","abstract_canon_sha256":"429112aeb1d209a0febfc6d51f0d78ae7e2258c6af3714037e95db14a6b5bdf1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:23.317956Z","signature_b64":"sMkIsoYm2OkdKRY5hwjOfyQdcYx5F+PqQJvcpcVIzJbZQAdefvHAl+9I086ZKahnJ8i5cCQmFc1OGvEJYluXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c4eb8e6f01b571417b300e71721505030ccd0cf552dde25f5dadab50786e662","last_reissued_at":"2026-05-18T02:29:23.317469Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:23.317469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analysis of the archetypal functional equation in the non-critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Gregory Derfel, Leonid V. 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