{"paper":{"title":"The Lebesgue Constant for the Periodic Franklin System","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Markus Passenbrunner","submitted_at":"2011-03-10T07:02:03Z","abstract_excerpt":"We identify the torus with the unit interval $[0,1)$ and let $n,\\nu\\in\\mathbb{N}$, $1\\leq \\nu\\leq n-1$ and $N:=n+\\nu$. Then we define the (partially equally spaced) knots \\[ t_{j}=\\{[c]{ll}% \\frac{j}{2n}, & \\text{for}j=0,...,2\\nu, \\frac{j-\\nu}{n}, & \\text{for}j=2\\nu+1,...,N-1.] Furthermore, given $n,\\nu$ we let $V_{n,\\nu}$ be the space of piecewise linear continuous functions on the torus with knots $\\{t_j:0\\leq j\\leq N-1\\}$. Finally, let $P_{n,\\nu}$ be the orthogonal projection operator of $L^{2}([0,1))$ onto $V_{n,\\nu}.$ The main result is \\[\\lim_{n\\rightarrow\\infty,\\nu=1}\\|P_{n,\\nu}:L^\\inft"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1950","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"}