{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:WWBND75ZAZFCFFIOB6TZWAXTB3","short_pith_number":"pith:WWBND75Z","canonical_record":{"source":{"id":"1707.03917","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-12T21:24:05Z","cross_cats_sorted":["math.AP","math.SP"],"title_canon_sha256":"6e91a92987abf0adf953ba110dcd8bd6aecc3d8b2986252b48efbbce02a97552","abstract_canon_sha256":"7aa321db8b60f05672b17ab51eb30eb41526f2b30d4a6707a4a815638f96af9d"},"schema_version":"1.0"},"canonical_sha256":"b582d1ffb9064a22950e0fa79b02f30ef189166967e32c275cac556e225aef83","source":{"kind":"arxiv","id":"1707.03917","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.03917","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"arxiv_version","alias_value":"1707.03917v1","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.03917","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"pith_short_12","alias_value":"WWBND75ZAZFC","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WWBND75ZAZFCFFIO","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WWBND75Z","created_at":"2026-05-18T12:31:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:WWBND75ZAZFCFFIOB6TZWAXTB3","target":"record","payload":{"canonical_record":{"source":{"id":"1707.03917","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-12T21:24:05Z","cross_cats_sorted":["math.AP","math.SP"],"title_canon_sha256":"6e91a92987abf0adf953ba110dcd8bd6aecc3d8b2986252b48efbbce02a97552","abstract_canon_sha256":"7aa321db8b60f05672b17ab51eb30eb41526f2b30d4a6707a4a815638f96af9d"},"schema_version":"1.0"},"canonical_sha256":"b582d1ffb9064a22950e0fa79b02f30ef189166967e32c275cac556e225aef83","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:23.134974Z","signature_b64":"Hwph6GiQNS9ZJvro+ppXqkP4tIkUPYX1KHKmFN+G4eAJ4kkG9oHnJIaEmPBkvywoHHGv9+EkHUkptoSojlBICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b582d1ffb9064a22950e0fa79b02f30ef189166967e32c275cac556e225aef83","last_reissued_at":"2026-05-18T00:40:23.134195Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:23.134195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.03917","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":"2026-05-18T00:40:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D3DhKqNACPgjLPWf5BYJ7t8hn/B30zyPv9KbL9sRwfVMXFLOsjexITnG1Y75oxwi7qq5yX1VDov2FUtt0/itBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:09:24.549834Z"},"content_sha256":"7debd4a504defae51989d71b1951f8635ce82ed647e455d634805145b0e140c9","schema_version":"1.0","event_id":"sha256:7debd4a504defae51989d71b1951f8635ce82ed647e455d634805145b0e140c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:WWBND75ZAZFCFFIOB6TZWAXTB3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.SP"],"primary_cat":"math.FA","authors_text":"Aparajita Dasgupta, Michael Ruzhansky","submitted_at":"2017-07-12T21:24:05Z","abstract_excerpt":"In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mapp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03917","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":"2026-05-18T00:40:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pAP1VHMewWHaxvDyncmfhQyjroCbtElKmJ4x8kTr4lL4sde8r+il0eNVAXxXgYz/G8Yrg6MjarqkxozmJlSOBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:09:24.550605Z"},"content_sha256":"dd6c5fb64c000d0ad39949329bd7580cf45ca0de3f3528665e30eca032472974","schema_version":"1.0","event_id":"sha256:dd6c5fb64c000d0ad39949329bd7580cf45ca0de3f3528665e30eca032472974"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WWBND75ZAZFCFFIOB6TZWAXTB3/bundle.json","state_url":"https://pith.science/pith/WWBND75ZAZFCFFIOB6TZWAXTB3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WWBND75ZAZFCFFIOB6TZWAXTB3/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":"2026-06-09T19:09:24Z","links":{"resolver":"https://pith.science/pith/WWBND75ZAZFCFFIOB6TZWAXTB3","bundle":"https://pith.science/pith/WWBND75ZAZFCFFIOB6TZWAXTB3/bundle.json","state":"https://pith.science/pith/WWBND75ZAZFCFFIOB6TZWAXTB3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WWBND75ZAZFCFFIOB6TZWAXTB3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:WWBND75ZAZFCFFIOB6TZWAXTB3","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":"7aa321db8b60f05672b17ab51eb30eb41526f2b30d4a6707a4a815638f96af9d","cross_cats_sorted":["math.AP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-12T21:24:05Z","title_canon_sha256":"6e91a92987abf0adf953ba110dcd8bd6aecc3d8b2986252b48efbbce02a97552"},"schema_version":"1.0","source":{"id":"1707.03917","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.03917","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"arxiv_version","alias_value":"1707.03917v1","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.03917","created_at":"2026-05-18T00:40:23Z"},{"alias_kind":"pith_short_12","alias_value":"WWBND75ZAZFC","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"WWBND75ZAZFCFFIO","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"WWBND75Z","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:dd6c5fb64c000d0ad39949329bd7580cf45ca0de3f3528665e30eca032472974","target":"graph","created_at":"2026-05-18T00:40:23Z","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":"In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mapp","authors_text":"Aparajita Dasgupta, Michael Ruzhansky","cross_cats":["math.AP","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-12T21:24:05Z","title":"Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03917","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":"sha256:7debd4a504defae51989d71b1951f8635ce82ed647e455d634805145b0e140c9","target":"record","created_at":"2026-05-18T00:40:23Z","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":"7aa321db8b60f05672b17ab51eb30eb41526f2b30d4a6707a4a815638f96af9d","cross_cats_sorted":["math.AP","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-12T21:24:05Z","title_canon_sha256":"6e91a92987abf0adf953ba110dcd8bd6aecc3d8b2986252b48efbbce02a97552"},"schema_version":"1.0","source":{"id":"1707.03917","kind":"arxiv","version":1}},"canonical_sha256":"b582d1ffb9064a22950e0fa79b02f30ef189166967e32c275cac556e225aef83","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b582d1ffb9064a22950e0fa79b02f30ef189166967e32c275cac556e225aef83","first_computed_at":"2026-05-18T00:40:23.134195Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:23.134195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Hwph6GiQNS9ZJvro+ppXqkP4tIkUPYX1KHKmFN+G4eAJ4kkG9oHnJIaEmPBkvywoHHGv9+EkHUkptoSojlBICw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:23.134974Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.03917","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7debd4a504defae51989d71b1951f8635ce82ed647e455d634805145b0e140c9","sha256:dd6c5fb64c000d0ad39949329bd7580cf45ca0de3f3528665e30eca032472974"],"state_sha256":"d835cbc67981aac0c869b2c0706b272a6dcde208f06fe73f317c2dbeae2810c5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LdpSZO9Scy0lAVeUuTbZMqQcnV+ee3PFCv9kNNIOQeiBabXMjxX23c3VjOQS4eO+xnmp5pkF+7rOV+BIPnDQCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T19:09:24.554852Z","bundle_sha256":"83ae1805c07d6f8aca079c41f55df61d33c38bde46f892c913ceb8a750b456df"}}