{"paper":{"title":"On almost everywhere convergence of tensor product spline projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Joscha Prochno, Markus Passenbrunner","submitted_at":"2013-10-24T07:16:48Z","abstract_excerpt":"Let $d\\in\\mathbb N$ and $f$ be a function in the Orlicz class $L(\\log^+L)^{d-1}$ defined on the unit cube $[0,1]^d$ in $\\mathbb{R}^d$. Given partitions $\\Delta_1,\\ldots,$ $\\Delta_d$ of $[0,1]$, we first prove that the orthogonal projection $P_{(\\Delta_1,\\dots,\\Delta_d)}(f)$ onto the space of tensor product splines with arbitrary orders $(k_1,\\dots, k_d)$ and knots $\\Delta_1,\\ldots,\\Delta_d$ converges to $f$ almost everywhere as the mesh diameters $|\\Delta_1|,\\ldots, |\\Delta_{d}|$ tend to zero. This extends the one-dimensional result in [Passenbrunner and Shadrin, Journal of Approximation Theor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.6505","kind":"arxiv","version":5},"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"}