{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TXOX5JDVATE4XKHW43R5QYLCN2","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":"d0d7f59bc90b737b69f627f77b52511b13d872bba4a8a3843877fcd0f0d7d898","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-07-27T08:51:51Z","title_canon_sha256":"95852994e4250e55600175382e729711fc2ef8e5381d98c37897145b94e26b47"},"schema_version":"1.0","source":{"id":"1107.5415","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.5415","created_at":"2026-05-18T03:20:57Z"},{"alias_kind":"arxiv_version","alias_value":"1107.5415v2","created_at":"2026-05-18T03:20:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.5415","created_at":"2026-05-18T03:20:57Z"},{"alias_kind":"pith_short_12","alias_value":"TXOX5JDVATE4","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TXOX5JDVATE4XKHW","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TXOX5JDV","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:35196278451dd654fbedc424ed45fb292da4862b6cffd3fc30129f0833659020","target":"graph","created_at":"2026-05-18T03:20:57Z","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":"We introduce a fast Fourier transform on regular d-dimensional lattices. We investigate properties of congruence class representants, i.e. their ordering, to classify directions and derive a Cooley-Tukey-Algorithm. Despite the fast Fourier techniques itself, there is also the advantage of this transform to be parallelized efficiently, yielding faster versions than the one-dimensional Fourier transform. These properties of the lattice can further be used to perform a fast multivariate wavelet decomposition, where the wavelets are given as trigonometric polynomials. Furthermore the preferred dir","authors_text":"Ronny Bergmann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-07-27T08:51:51Z","title":"The fast Fourier Transform and fast Wavelet Transform for Patterns on the Torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.5415","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":"sha256:2f4f5e93c70694e6b3a1f232892c14b3aaea66cf784711be88cc112cba3c4f1b","target":"record","created_at":"2026-05-18T03:20:57Z","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":"d0d7f59bc90b737b69f627f77b52511b13d872bba4a8a3843877fcd0f0d7d898","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2011-07-27T08:51:51Z","title_canon_sha256":"95852994e4250e55600175382e729711fc2ef8e5381d98c37897145b94e26b47"},"schema_version":"1.0","source":{"id":"1107.5415","kind":"arxiv","version":2}},"canonical_sha256":"9ddd7ea47504c9cba8f6e6e3d861626ebe4440f23e2b5d2d4c791237dc7aef7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ddd7ea47504c9cba8f6e6e3d861626ebe4440f23e2b5d2d4c791237dc7aef7f","first_computed_at":"2026-05-18T03:20:57.161080Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:20:57.161080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wOdoJPmUHhkllNLTm3CNq/qiTAwyX+khKADjSf4VfXFx5hO/uhO2I0AN/L5hWzUtFTIYrgvbgpRZeEXB+5R/Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:20:57.161824Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.5415","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f4f5e93c70694e6b3a1f232892c14b3aaea66cf784711be88cc112cba3c4f1b","sha256:35196278451dd654fbedc424ed45fb292da4862b6cffd3fc30129f0833659020"],"state_sha256":"5a2ba7232c0aaeb56afa922793b3a6cca775f1f44d035aa915b22dfc14c12e04"}