{"paper":{"title":"Bracket products for Weyl-Heisenberg frames","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"M. C. Lammers, Peter G. Casazza","submitted_at":"1999-11-04T18:06:06Z","abstract_excerpt":"We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor,\n Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to\n Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel's inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions $(g_{n})$ into a sequence $(e_{n})$ with the property that $(E_{mb}e_{n})_{m,n\\in \\Bbb Z}$ is orthonormal in $L^{2}(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9911026","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"}