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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"},"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":"1903.07974","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-03-19T13:04:46Z","cross_cats_sorted":[],"title_canon_sha256":"42d77c7ce0cac4f85f1016c559e62fb4a112b41599161b25e1a307adc9f1344d","abstract_canon_sha256":"46737721900c923c21385e79380a4452ef5d07913d653330e3ab1f1a853bd1d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:53.344409Z","signature_b64":"HHzJwOt2M/g46qzkIHLz4GZMH5MPIVedbp1q/xXGH/Jre1+e6fXGcCIz9s/y9h1RfSvd660Dh8S0TvpIhQkGBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03ae6d97c64a9fe5a84099719097ec7353ede463638d8e660de336749a065816","last_reissued_at":"2026-05-17T23:50:53.343223Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:53.343223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1903.07974","created_at":"2026-05-17T23:50:53.343375+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.07974v1","created_at":"2026-05-17T23:50:53.343375+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.07974","created_at":"2026-05-17T23:50:53.343375+00:00"},{"alias_kind":"pith_short_12","alias_value":"AOXG3F6GJKP6","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"AOXG3F6GJKP6LKCA","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"AOXG3F6G","created_at":"2026-05-18T12:33:12.712433+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON","json":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON.json","graph_json":"https://pith.science/api/pith-number/AOXG3F6GJKP6LKCATFYZBF7MON/graph.json","events_json":"https://pith.science/api/pith-number/AOXG3F6GJKP6LKCATFYZBF7MON/events.json","paper":"https://pith.science/paper/AOXG3F6G"},"agent_actions":{"view_html":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON","download_json":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON.json","view_paper":"https://pith.science/paper/AOXG3F6G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.07974&json=true","fetch_graph":"https://pith.science/api/pith-number/AOXG3F6GJKP6LKCATFYZBF7MON/graph.json","fetch_events":"https://pith.science/api/pith-number/AOXG3F6GJKP6LKCATFYZBF7MON/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON/action/storage_attestation","attest_author":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON/action/author_attestation","sign_citation":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON/action/citation_signature","submit_replication":"https://pith.science/pith/AOXG3F6GJKP6LKCATFYZBF7MON/action/replication_record"}},"created_at":"2026-05-17T23:50:53.343375+00:00","updated_at":"2026-05-17T23:50:53.343375+00:00"}