{"paper":{"title":"The Fourier transform of order statistics with applications to Lorentz spaces","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexander Koldobsky, Stephen J. Dilworth","submitted_at":"1993-11-17T18:36:11Z","abstract_excerpt":"We present a formula for the Fourier transforms of order statistics in $\\Bbb R^n$ showing that all these Fourier transforms are equal up to a constant multiple outside the coordinate planes in $\\Bbb R^n.$\n  For $a_1\\geq ... \\geq a_n\\ge0$ and $q>0,$ denote by $\\ell_{w,q}^n$ the $n$-dimensional Lorentz space with the norm $\\|(x_1,...,x_n)\\| = (a_1 (x_1^{*})^q +...+ a_n (x_n^{*})^q)^{1/q}$, where $(x_1^{*},...,x_n^{*})$ is the non-increasing permutation of the numbers $|x_1|,...,|x_n|.$ We use the above mentioned formula and the Fourier transform criterion of isometric embeddability of Banach spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9311208","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"}