{"paper":{"title":"Fr\\'echet frames, general definition and expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Diana T. Stoeva, Stevan Pilipovi\\'c","submitted_at":"2012-01-10T16:11:08Z","abstract_excerpt":"We define an {\\it $(X_1,\\Theta, X_2)$-frame} with Banach spaces $X_2\\subseteq X_1$, $|\\cdot|_1 \\leq |\\cdot|_2$, and a $BK$-space $(\\Theta, \\snorm[\\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\\infty$ and of sequence spaces ${\\Theta_s}_{s=0}^\\infty$, we define a general Fr\\' echet frame on the Fr\\' echet space $X_F=\\bigcap_{s=0}^\\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.2096","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"}