{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3TNGPBPSNJJZPDEIVUYEP6YKN6","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":"2af878fe84bc8f3ff70b31e44987c1c85a8bc6e7eb3b76e5f360f0f1e6b21ee9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-17T14:03:51Z","title_canon_sha256":"cfc2a02cbfc374b99644f830b7f2931a39b969590465032f8726430e8d65f473"},"schema_version":"1.0","source":{"id":"1605.05168","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.05168","created_at":"2026-05-18T00:27:47Z"},{"alias_kind":"arxiv_version","alias_value":"1605.05168v2","created_at":"2026-05-18T00:27:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.05168","created_at":"2026-05-18T00:27:47Z"},{"alias_kind":"pith_short_12","alias_value":"3TNGPBPSNJJZ","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"3TNGPBPSNJJZPDEI","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"3TNGPBPS","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:29f1d345eeeb587a264303511ae1fcc86a4d5d09c6850918ecf03aa990210d7b","target":"graph","created_at":"2026-05-18T00:27:47Z","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":"The Zak transform on $\\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak transform to a class of locally compact $G$-spaces, where $G$ is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak transform. It is shown that the Zak transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of $L^2$-function","authors_text":"Dominik J\\\"ustel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-17T14:03:51Z","title":"The Zak transform on strongly proper $G$-spaces and its applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05168","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":"sha256:910d1f80b54937fec787530bb6c99cb4bab680d058ec9b52f7ffe719fbb7789a","target":"record","created_at":"2026-05-18T00:27:47Z","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":"2af878fe84bc8f3ff70b31e44987c1c85a8bc6e7eb3b76e5f360f0f1e6b21ee9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-17T14:03:51Z","title_canon_sha256":"cfc2a02cbfc374b99644f830b7f2931a39b969590465032f8726430e8d65f473"},"schema_version":"1.0","source":{"id":"1605.05168","kind":"arxiv","version":2}},"canonical_sha256":"dcda6785f26a53978c88ad3047fb0a6fa5f77229cdc20e4a9ad6f8ab47347b7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dcda6785f26a53978c88ad3047fb0a6fa5f77229cdc20e4a9ad6f8ab47347b7a","first_computed_at":"2026-05-18T00:27:47.235446Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:27:47.235446Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GlQTjwxrjSSmWPwJ+ASOCIEjhxoXrX/elcyBIF6OR0hlS2dmdmpa1ZMRxksJU4HCppmhDX0CtR/1OqhO2fxDCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:27:47.235921Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.05168","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:910d1f80b54937fec787530bb6c99cb4bab680d058ec9b52f7ffe719fbb7789a","sha256:29f1d345eeeb587a264303511ae1fcc86a4d5d09c6850918ecf03aa990210d7b"],"state_sha256":"c48b60226ba40c81dbf3cbdbcddd92e961e8c48ef1774a5ad70df78ebb829e88"}