{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:KBA4ZBKV7HAAMBOZTH7OVVN357","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c6f8d0ab3f0c8f86142f4efa771a24a619a0c880733dd208bc9b75e309c9d9f4","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-01-05T14:36:31Z","title_canon_sha256":"0922d44b35567725c72f0a319939f110ce9ce341e7e02ac33ad2ddf0359681e5"},"schema_version":"1.0","source":{"id":"1201.1179","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.1179","created_at":"2026-05-18T02:58:22Z"},{"alias_kind":"arxiv_version","alias_value":"1201.1179v2","created_at":"2026-05-18T02:58:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1179","created_at":"2026-05-18T02:58:22Z"},{"alias_kind":"pith_short_12","alias_value":"KBA4ZBKV7HAA","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"KBA4ZBKV7HAAMBOZ","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"KBA4ZBKV","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:459767d72b0688c5132eaa1d1ad85b8997cc4560f2bd48cf3a5bb09eef42f3e3","target":"graph","created_at":"2026-05-18T02:58:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $H$ and $K$ be locally compact groups and also $\\tau:H\\to Aut(K)$ be a continuous homomorphism and $G_\\tau=H\\ltimes_\\tau K$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $\\tau$. This paper presents a novel approach to the Fourier analysis of $G_\\tau$, when $K$ is abelian. We define the $\\tau$-dual group $G_{\\hat{\\tau}}$ of $G_\\tau$ as the semi-direct product $H\\ltimes_{\\hat{\\tau}}\\hat{K}$, where $\\hat{\\tau}:H\\to Aut(\\hat{K})$ defined via (\\ref{A}). We prove a Ponterjagin duality Theorem and also we study $\\tau$-Fourier transforms on $G_\\tau$. As a co","authors_text":"Arash Ghaani Farashahi","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-01-05T14:36:31Z","title":"A new approach to the Fourier analysis on semi-direct products of groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1179","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d0d79b12c1ae0698e0e79d3bfd3781a0ac38271b4259692d8fefdad302a661be","target":"record","created_at":"2026-05-18T02:58:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c6f8d0ab3f0c8f86142f4efa771a24a619a0c880733dd208bc9b75e309c9d9f4","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-01-05T14:36:31Z","title_canon_sha256":"0922d44b35567725c72f0a319939f110ce9ce341e7e02ac33ad2ddf0359681e5"},"schema_version":"1.0","source":{"id":"1201.1179","kind":"arxiv","version":2}},"canonical_sha256":"5041cc8555f9c00605d999feead5bbefc52b6c426e36053ab0b8928f5c58db42","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5041cc8555f9c00605d999feead5bbefc52b6c426e36053ab0b8928f5c58db42","first_computed_at":"2026-05-18T02:58:22.261465Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:22.261465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5gehD2OgTOYh1Z6Smpknq8V5Z+0zerDjsFQBbNdi7w3vDP8WtnOLjLnWb/C1/w2VWbO1hwRY/fi9xRWm6N8nAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:22.261901Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.1179","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0d79b12c1ae0698e0e79d3bfd3781a0ac38271b4259692d8fefdad302a661be","sha256:459767d72b0688c5132eaa1d1ad85b8997cc4560f2bd48cf3a5bb09eef42f3e3"],"state_sha256":"930477c82d007e4f5d4eb907223dc2e0e35b223d9c12b47413f15983f34197c7"}