{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:44WNKERCQ6H3TJQYHDJPCATF2Y","short_pith_number":"pith:44WNKERC","canonical_record":{"source":{"id":"1012.1801","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-12-08T16:45:06Z","cross_cats_sorted":[],"title_canon_sha256":"5ab95d65d3530d5694b9c732fe4efeae3c0b2022d7c6956ff3b4279abe171111","abstract_canon_sha256":"0e21d2569d629e8186ec2155988749f8a02370f546be5cd6f9d8db1729a64fab"},"schema_version":"1.0"},"canonical_sha256":"e72cd51222878fb9a61838d2f10265d636c9f60733ac40fd3da435e2f3d52012","source":{"kind":"arxiv","id":"1012.1801","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.1801","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"arxiv_version","alias_value":"1012.1801v1","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1801","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"pith_short_12","alias_value":"44WNKERCQ6H3","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"44WNKERCQ6H3TJQY","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"44WNKERC","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:44WNKERCQ6H3TJQYHDJPCATF2Y","target":"record","payload":{"canonical_record":{"source":{"id":"1012.1801","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-12-08T16:45:06Z","cross_cats_sorted":[],"title_canon_sha256":"5ab95d65d3530d5694b9c732fe4efeae3c0b2022d7c6956ff3b4279abe171111","abstract_canon_sha256":"0e21d2569d629e8186ec2155988749f8a02370f546be5cd6f9d8db1729a64fab"},"schema_version":"1.0"},"canonical_sha256":"e72cd51222878fb9a61838d2f10265d636c9f60733ac40fd3da435e2f3d52012","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:51.230990Z","signature_b64":"j0S2yh8z7c47X4u0DfG0JeIcnJnbO2F9g9TWCIaU4R05BioGvwJeyItZ7dnLioFtabBPnjC55jwvlpC06Wv/Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e72cd51222878fb9a61838d2f10265d636c9f60733ac40fd3da435e2f3d52012","last_reissued_at":"2026-05-18T04:33:51.230287Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:51.230287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.1801","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:33:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OIWlhn9bMJHKslWaHGP2IJ/Mad0rylIrtdUo28cUIZQzIVKkfGOUYwTLt+ZPqQLiZAPwndicqKlRblGcrJEEDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T07:37:49.519865Z"},"content_sha256":"7816142afbf1fc32a51e054d77642f5fca1a7b9625edaf67b734f040bcd0b772","schema_version":"1.0","event_id":"sha256:7816142afbf1fc32a51e054d77642f5fca1a7b9625edaf67b734f040bcd0b772"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:44WNKERCQ6H3TJQYHDJPCATF2Y","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Paley-Wiener Theorems with respect to the spectral parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Gestur Olafsson, Susanna Dann","submitted_at":"2010-12-08T16:45:06Z","abstract_excerpt":"One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic analysis is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations. We give a short overview of the Fourier transform on such spaces and then ask if one can describe the image of the space of smooth compactly supported functions in terms of the spectral parameter, i.e., the parameterization of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1801","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:33:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yxch5SGiNrpy926xErwHlgjZ2zD20bYAJyKDGcpHtYFs6baY0SyAoRzpNJhc8xhs2yvlM1P85a+HuUn0ULYPCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T07:37:49.520473Z"},"content_sha256":"fcb8a2d534757792f4c3a6a0859951c913badffd6eb3aed400f76a695b75e45c","schema_version":"1.0","event_id":"sha256:fcb8a2d534757792f4c3a6a0859951c913badffd6eb3aed400f76a695b75e45c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/44WNKERCQ6H3TJQYHDJPCATF2Y/bundle.json","state_url":"https://pith.science/pith/44WNKERCQ6H3TJQYHDJPCATF2Y/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/44WNKERCQ6H3TJQYHDJPCATF2Y/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-21T07:37:49Z","links":{"resolver":"https://pith.science/pith/44WNKERCQ6H3TJQYHDJPCATF2Y","bundle":"https://pith.science/pith/44WNKERCQ6H3TJQYHDJPCATF2Y/bundle.json","state":"https://pith.science/pith/44WNKERCQ6H3TJQYHDJPCATF2Y/state.json","well_known_bundle":"https://pith.science/.well-known/pith/44WNKERCQ6H3TJQYHDJPCATF2Y/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:44WNKERCQ6H3TJQYHDJPCATF2Y","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":"0e21d2569d629e8186ec2155988749f8a02370f546be5cd6f9d8db1729a64fab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-12-08T16:45:06Z","title_canon_sha256":"5ab95d65d3530d5694b9c732fe4efeae3c0b2022d7c6956ff3b4279abe171111"},"schema_version":"1.0","source":{"id":"1012.1801","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.1801","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"arxiv_version","alias_value":"1012.1801v1","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1801","created_at":"2026-05-18T04:33:51Z"},{"alias_kind":"pith_short_12","alias_value":"44WNKERCQ6H3","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"44WNKERCQ6H3TJQY","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"44WNKERC","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:fcb8a2d534757792f4c3a6a0859951c913badffd6eb3aed400f76a695b75e45c","target":"graph","created_at":"2026-05-18T04:33:51Z","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":"One of the important questions related to any integral transform on a manifold M or on a homogeneous space G/K is the description of the image of a given space of functions. If M=G/K, where (G,K) is a Gelfand pair, then the harmonic analysis is closely related to the representations of G and the direct integral decomposition of L^2(M) into irreducible representations. We give a short overview of the Fourier transform on such spaces and then ask if one can describe the image of the space of smooth compactly supported functions in terms of the spectral parameter, i.e., the parameterization of th","authors_text":"Gestur Olafsson, Susanna Dann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-12-08T16:45:06Z","title":"Paley-Wiener Theorems with respect to the spectral parameter"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1801","kind":"arxiv","version":1},"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":"sha256:7816142afbf1fc32a51e054d77642f5fca1a7b9625edaf67b734f040bcd0b772","target":"record","created_at":"2026-05-18T04:33:51Z","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":"0e21d2569d629e8186ec2155988749f8a02370f546be5cd6f9d8db1729a64fab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-12-08T16:45:06Z","title_canon_sha256":"5ab95d65d3530d5694b9c732fe4efeae3c0b2022d7c6956ff3b4279abe171111"},"schema_version":"1.0","source":{"id":"1012.1801","kind":"arxiv","version":1}},"canonical_sha256":"e72cd51222878fb9a61838d2f10265d636c9f60733ac40fd3da435e2f3d52012","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e72cd51222878fb9a61838d2f10265d636c9f60733ac40fd3da435e2f3d52012","first_computed_at":"2026-05-18T04:33:51.230287Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:51.230287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j0S2yh8z7c47X4u0DfG0JeIcnJnbO2F9g9TWCIaU4R05BioGvwJeyItZ7dnLioFtabBPnjC55jwvlpC06Wv/Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:51.230990Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.1801","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7816142afbf1fc32a51e054d77642f5fca1a7b9625edaf67b734f040bcd0b772","sha256:fcb8a2d534757792f4c3a6a0859951c913badffd6eb3aed400f76a695b75e45c"],"state_sha256":"de3a08491768f426644380edd7285484edd3fe506c2c7a962f79ff41f30fceea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dy++CrMCIEVssNvD3RnoSzQBUs4y+t40S0oBAeEyA3u0VjUa8FKLtR/snPrGQxi1IB7HW01HMgmbA3AWepZXCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T07:37:49.522631Z","bundle_sha256":"00ead03df7bc2575ceafa336cf28c3b8a1c55a7b76f645e67a8618c8fa97e2da"}}