{"paper":{"title":"On the Sign Distributions of Hilbert Space Frames","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Alexander Volberg, Nikolai Nikolski","submitted_at":"2018-12-15T16:15:00Z","abstract_excerpt":"We show that the positive and negative parts $ u_{k}^{\\pm }$ of any frame in a real $ L^{2}$ space with respect to a continuous measure have both \"infinite $ l^{2}$ masses\": 1) always, $ \\sum _{k}u_{k}^{\\pm }(x)^{2}=\\infty $ almost everywhere (in particular, there exist no positive frames, nor Riesz bases), but 2) $ \\sum _{k=1}^{n}(u_{k}^{+}(x)-u_{k}^{-}(x))^{2}$ can grow \"locally\" as slow as we wish (for $ n\\longrightarrow \\infty $), and 3) it can happen that $ \\sum _{k=1}^{n}u_{k}^{-}(x)^{2}=\\, o(\\sum _{k=1}^{n}u_{k}^{+}(x)^{2})$, and vice versa, as $ n\\longrightarrow \\infty $ on a set of po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.06313","kind":"arxiv","version":3},"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"}