{"paper":{"title":"Martingale inequalities for spline sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Markus Passenbrunner","submitted_at":"2018-12-19T08:54:10Z","abstract_excerpt":"We show that D. L\\'{e}pingle's $L_1(\\ell_2)$-inequality \\begin{equation*}\n  \\Big\\| \\big( \\sum_n \\mathbb E[f_n | \\mathscr F_{n-1}]^2\n  \\big)^{1/2}\\Big\\|_1 \\leq 2\\cdot \\Big\\| \\big( \\sum_n f_n^2\n  \\big)^{1/2} \\Big\\|_1, \\qquad f_n\\in\\mathscr F_n,\n  \\end{equation*} extends to the case where we substitute the conditional expectation operators with orthogonal projection operators onto spline spaces and where we can allow that $f_n$ is contained in a suitable spline space $\\mathscr S(\\mathscr F_n)$. This is done provided the filtration $(\\mathscr F_n)$ satisfies a certain regularity condition dependin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.07817","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"}