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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/9311208","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1993-11-17T18:36:11Z","cross_cats_sorted":[],"title_canon_sha256":"30f299e54d1dbf395be6c0d13ad9630adbc2142802b2daa030d2e3a77b11e84a","abstract_canon_sha256":"50e16b87b67b45bb87cb5cd3bbff88ec752d1417d518619b83d8e06fc9914f5b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.799403Z","signature_b64":"HxobHzHmYTHALF1yPYtiN17fqt18IugwItyFVcdhGIr2dWF7ibIfgAwWhXY3G+JWRQq3T/SQSWV7eU7fQ6NmAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"530da8426d8d1aeb3216958959c7a87bfa49ea4b5ab15aa898746afd466ad92d","last_reissued_at":"2026-05-18T01:05:51.798987Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.798987Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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