{"paper":{"title":"Discrete Frames For $L^2({\\mathbb R}^{n^2})$ Arising From Tiling Systems On ${\\rm GL}_n({\\mathbb R})$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Kris Hollingsworth, Mahya Ghandehari","submitted_at":"2020-03-29T19:07:35Z","abstract_excerpt":"A discrete frame for $L^2({\\mathbb R}^d)$ is a countable sequence $\\{e_j\\}_{j\\in J}$ in $L^2({\\mathbb R}^d)$ together with real constants $0<A\\leq B< \\infty$ such that $$ A\\|f\\|_2^2 \\leq \\sum_{j\\in J}|\\langle f,e_j \\rangle |^2 \\leq B\\|f\\|_2^2,$$ for all $f\\in L^2(\\mathbb{R}^d)$. We present a method of sampling continuous frames, which arise from square-integrable representations of affine-type groups, to create discrete frames for high-dimensional signals. Our method relies on partitioning the ambient space by using a suitable \"tiling system\". We provide all relevant details for constructions "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2003.13113","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2003.13113/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}