{"paper":{"title":"On almost everywhere convergence of orthogonal spline projections with arbitrary knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexei Shadrin, Markus Passenbrunner","submitted_at":"2013-08-22T11:26:49Z","abstract_excerpt":"The main result of this paper is a proof that, for any $f \\in L_1[a,b]$, a sequence of its orthogonal projections $(P_{\\Delta_n}(f))$ onto splines of order $k$ with arbitrary knots $\\Delta_n$, converges almost everywhere provided that the mesh diameter $|\\Delta_n|$ tends to zero, namely \\[ f \\in L_1[a,b] \\Rightarrow P_{\\Delta_n}(f,x) \\to f(x) \\quad \\mbox{a.e.} \\quad (|\\Delta_n|\\to 0)\\,. \\] This extends the earlier result that, for $f \\in L_p$, we have convergence $P_{\\Delta_n}(f) \\to f$ in the $L_p$-norm for $1 \\le p \\le \\infty$.}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4824","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"}