{"paper":{"title":"Tight frame completions with prescribed norms","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"M. Ruiz, P. Massey","submitted_at":"2006-06-13T21:08:21Z","abstract_excerpt":"Let $\\hil$ be a finite dimensional (real or complex) Hilbert space and let $\\{a_i\\}_{i=1}^\\infty$ be a non-increasing sequence of positive numbers. Given a finite sequence of vectors $\\f$ in $\\hil$ we find necessary and sufficient conditions for the existence of\n $r\\in \\NN\\cup\\{\\infty\\}$ and a Bessel sequence $\\g$ in $\\hil$ such that $\\cF\\cup\\cG$ is a tight frame for $\\hil$ and $\\|g_i\\|^2=a_i$ for $1\\leq i\\leq r$. Moreover, in this case we compute the minimum $r\\in \\NN\\cup\\{\\infty\\}$ with this property. Using recent results on the Schur-Horn theorem, we also obtain a not so optimal but algorit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0606319","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"}