{"paper":{"title":"Complex best $r$-term approximations almost always exist in finite dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NA","authors_text":"Lek-Heng Lim, Mateusz Micha{\\l}ek, Yang Qi","submitted_at":"2017-11-30T08:53:37Z","abstract_excerpt":"We show that in finite-dimensional nonlinear approximations, the best $r$-term approximant of a function $f$ almost always exists over $\\mathbb{C}$ but that the same is not true over $\\mathbb{R}$, i.e., the infimum $\\inf_{f_1,\\dots,f_r \\in Y} \\lVert f - f_1 - \\dots - f_r \\rVert$ is almost always attainable by complex-valued functions $f_1,\\dots, f_r$ in $Y$, a set of functions that have some desired structures. Our result extends to functions that possess special properties like symmetry or skew-symmetry under permutations of arguments. For the case where $Y$ is the set of separable functions,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11269","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"}