{"paper":{"title":"A new realization of rational functions, with applications to linear combination interpolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Daniel Alpay, Dan Volok, Izchak Lewkowicz, Palle Jorgensen","submitted_at":"2014-08-19T17:28:24Z","abstract_excerpt":"We introduce the following linear combination interpolation problem (LCI): Given $N$ distinct numbers $w_1,\\ldots w_N$ and $N+1$ complex numbers $a_1,\\ldots, a_N$ and $c$, find all functions $f(z)$ analytic in a simply connected set (depending on $f$) containing the points $w_1,\\ldots,w_N$ such that \\[ \\sum_{u=1}^Na_uf(w_u)=c. \\] To this end we prove a representation theorem for such functions $f$ in terms of an associated polynomial $p(z)$. We first introduce the following two operations, $(i)$ substitution of $p$, and $(ii)$ multiplication by monomials $z^j, 0\\le j < N$. Then let $M$ be the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4404","kind":"arxiv","version":2},"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"}