{"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. We show that the formula $Ph=\\sum_{n=1}^{N}g_n\\!\\cdot"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06275","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"}