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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"},"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":"1507.06445","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-23T11:08:38Z","cross_cats_sorted":[],"title_canon_sha256":"5661f49760ff8b367341232d35a31b1268cfd26c9a587babdacc2ec243182890","abstract_canon_sha256":"818acf039594b92d7f213384948d7fa7f4034dbbd40ad2e0c1afd3013c0f8d83"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:24.949936Z","signature_b64":"MoxT6y9g8vjqPW+hhx+iH8n8Jt56t5c3wPlPOKcRB3DO12mC3gN3lTyteppMsNc83ajhvPYoQOceJCejo4eVAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdac5d208e471f5d1db683d0b9ec29cf790c37fc94bda29ca42332aee532eaac","last_reissued_at":"2026-05-18T01:36:24.949330Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:24.949330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1507.06445","created_at":"2026-05-18T01:36:24.949408+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.06445v1","created_at":"2026-05-18T01:36:24.949408+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.06445","created_at":"2026-05-18T01:36:24.949408+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZWWF2IEOI4PV","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZWWF2IEOI4PV2HNW","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZWWF2IEO","created_at":"2026-05-18T12:29:52.810259+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5","json":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5.json","graph_json":"https://pith.science/api/pith-number/ZWWF2IEOI4PV2HNWQPILT3BJZ5/graph.json","events_json":"https://pith.science/api/pith-number/ZWWF2IEOI4PV2HNWQPILT3BJZ5/events.json","paper":"https://pith.science/paper/ZWWF2IEO"},"agent_actions":{"view_html":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5","download_json":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5.json","view_paper":"https://pith.science/paper/ZWWF2IEO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.06445&json=true","fetch_graph":"https://pith.science/api/pith-number/ZWWF2IEOI4PV2HNWQPILT3BJZ5/graph.json","fetch_events":"https://pith.science/api/pith-number/ZWWF2IEOI4PV2HNWQPILT3BJZ5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5/action/storage_attestation","attest_author":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5/action/author_attestation","sign_citation":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5/action/citation_signature","submit_replication":"https://pith.science/pith/ZWWF2IEOI4PV2HNWQPILT3BJZ5/action/replication_record"}},"created_at":"2026-05-18T01:36:24.949408+00:00","updated_at":"2026-05-18T01:36:24.949408+00:00"}