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We show that the formula $Ph=\\sum_{n=1}^{N}g_n\\!\\cdot"},"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":"1811.06275","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-11-15T10:24:09Z","cross_cats_sorted":[],"title_canon_sha256":"712ea282d5004ad133230dcea53e4995d51e27323cd2a22f23feb4c17b433bfa","abstract_canon_sha256":"60edfd3f71c6502b6c7f52774570b6c8c9f66b773cce9210212834caa00f89e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:38.147091Z","signature_b64":"cddqoIu38Xi2TkH1619IeFMP7STrZE93XXq9GYeCfsfxW0VJgKtncuFCijPI4dJ0bBnNJTM5sk3/NYnYhBQVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8802a1682e8df6e04b6e868fd0b2770e39543ce2e17662c7f203ca2caf0a3c3","last_reissued_at":"2026-05-18T00:00:38.146619Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:38.146619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some Class of Linear Operators Involved in Functional Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janusz Morawiec, Thomas Z\\\"urcher","submitted_at":"2018-11-15T10:24:09Z","abstract_excerpt":"Fix $N\\in\\mathbb N$ and assume that for every $n\\in\\{1,\\ldots, N\\}$ the functions $f_n\\colon[0,1]\\to[0,1]$ and $g_n\\colon[0,1]\\to\\mathbb R$ are Lebesgue measurable, $f_n$ is almost everywhere approximately differentiable with $|g_n(x)|<|f'_n(x)|$ for almost all $x\\in [0,1]$, there exists $K\\in\\mathbb N$ such that the set $\\{x\\in [0,1]:\\mathrm{card}{f_n^{-1}(x)}>K\\}$ is of Lebesgue measure zero, $f_n$ satisfy Luzin's condition N, and the set $f_n^{-1}(A)$ is of Lebesgue measure zero for every set $A\\subset\\mathbb R$ of Lebesgue measure zero. 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