{"paper":{"title":"Pitt's inequalities and uncertainty principle for generalized Fourier transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dmitry Gorbachev, Sergey Tikhonov, Valery Ivanov","submitted_at":"2015-07-23T11:08:38Z","abstract_excerpt":"We study the two-parameter family of unitary operators \\[ \\mathcal{F}_{k,a}=\\exp\\Bigl(\\frac{i\\pi}{2a}\\,(2\\langle k\\rangle+{d}+a-2 )\\Bigr) \\exp\\Bigl(\\frac{i\\pi}{2a}\\,\\Delta_{k,a}\\Bigr), \\] which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator $\\Delta_{k,a}=|x|^{2-a}\\Delta_{k}-|x|^{a}$, $a>0$, where $\\Delta_{k}$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of $\\mathcal{F}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{\\lambda,a}$.\n  We obta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.06445","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"}