{"paper":{"title":"On a class of linear functional equations without range condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Csaba Vincze, Eszter Gselmann, Gergely Kiss","submitted_at":"2019-03-19T13:04:46Z","abstract_excerpt":"The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\\mathbb{K}$ and let $\\alpha_{i}, \\beta_{i}\\in \\mathbb{K}$, $i=1, \\ldots, n$ be arbitrarily fixed constants. We will describe all those functions $f, f_{i, j}\\colon X\\times Y\\to \\mathbb{K}$, $i, j=1, \\ldots, n$ that fulfill functional equation \\[ f\\left(\\sum_{i=1}^n \\alpha_i x_i, \\sum_{i=1}^n \\beta_i y_i\\right)= \\sum_{i, j=1}^{n}f_{i, j}(x_i, y_j) \\qquad \\left(x_i \\in X, y_i \\in Y, i=1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07974","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"}