{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:J6UOU3VXVXIAZFLOIPKKXQMP46","short_pith_number":"pith:J6UOU3VX","schema_version":"1.0","canonical_sha256":"4fa8ea6eb7add00c956e43d4abc18fe78e4fc742aaf20542b76ba873bba76c38","source":{"kind":"arxiv","id":"1203.1509","version":1},"attestation_state":"computed","paper":{"title":"Zak Transform for Semidirect Product of Locally Compact Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.FA","authors_text":"Ali Akbar Arefijamaal, Arash Ghaani Farashahi","submitted_at":"2012-03-07T15:56:10Z","abstract_excerpt":"Let $H$ be a locally compact group and $K$ be an LCA group also let $\\tau:H\\to Aut(K)$ be a continuous homomorphism and $G_\\tau=H\\ltimes_\\tau K$ be the semidirect product of $H$ and $K$ with respect to $\\tau$. In this article we define the Zak transform $\\mathcal{Z}_L$ on $L^2(G_\\tau)$ with respect to a $\\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\\mathrm{SL}(2,\\mathbb{Z})\\ltimes_\\tau\\mathbb{R}^2$ and also the Weyl-Heisenberg group"},"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":"1203.1509","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-03-07T15:56:10Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"8ee62c7e571f5677f8e98ac39cb69010f91bca13e9fa445f04149a2f9b8f2461","abstract_canon_sha256":"9e9fb49674cc5a580861bab654885c7b4c2873a9e73b0c51bd7f17442f978bcf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:38.567286Z","signature_b64":"AGc/tnEnfMD9hlrkkI5tWJaf1WSiNH2piWAeT3KPZd7fbnipDja7hq91sPZMULCamqWsV+ZM+85H1OD1WhKRDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fa8ea6eb7add00c956e43d4abc18fe78e4fc742aaf20542b76ba873bba76c38","last_reissued_at":"2026-05-18T04:00:38.566568Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:38.566568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zak Transform for Semidirect Product of Locally Compact Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.FA","authors_text":"Ali Akbar Arefijamaal, Arash Ghaani Farashahi","submitted_at":"2012-03-07T15:56:10Z","abstract_excerpt":"Let $H$ be a locally compact group and $K$ be an LCA group also let $\\tau:H\\to Aut(K)$ be a continuous homomorphism and $G_\\tau=H\\ltimes_\\tau K$ be the semidirect product of $H$ and $K$ with respect to $\\tau$. In this article we define the Zak transform $\\mathcal{Z}_L$ on $L^2(G_\\tau)$ with respect to a $\\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\\mathrm{SL}(2,\\mathbb{Z})\\ltimes_\\tau\\mathbb{R}^2$ and also the Weyl-Heisenberg group"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1509","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":"1203.1509","created_at":"2026-05-18T04:00:38.566691+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.1509v1","created_at":"2026-05-18T04:00:38.566691+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.1509","created_at":"2026-05-18T04:00:38.566691+00:00"},{"alias_kind":"pith_short_12","alias_value":"J6UOU3VXVXIA","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"J6UOU3VXVXIAZFLO","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"J6UOU3VX","created_at":"2026-05-18T12:27:11.947152+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/J6UOU3VXVXIAZFLOIPKKXQMP46","json":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46.json","graph_json":"https://pith.science/api/pith-number/J6UOU3VXVXIAZFLOIPKKXQMP46/graph.json","events_json":"https://pith.science/api/pith-number/J6UOU3VXVXIAZFLOIPKKXQMP46/events.json","paper":"https://pith.science/paper/J6UOU3VX"},"agent_actions":{"view_html":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46","download_json":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46.json","view_paper":"https://pith.science/paper/J6UOU3VX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.1509&json=true","fetch_graph":"https://pith.science/api/pith-number/J6UOU3VXVXIAZFLOIPKKXQMP46/graph.json","fetch_events":"https://pith.science/api/pith-number/J6UOU3VXVXIAZFLOIPKKXQMP46/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46/action/storage_attestation","attest_author":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46/action/author_attestation","sign_citation":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46/action/citation_signature","submit_replication":"https://pith.science/pith/J6UOU3VXVXIAZFLOIPKKXQMP46/action/replication_record"}},"created_at":"2026-05-18T04:00:38.566691+00:00","updated_at":"2026-05-18T04:00:38.566691+00:00"}