{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:25YPKIEQWR2YQM3KSRBNUTJCMJ","short_pith_number":"pith:25YPKIEQ","canonical_record":{"source":{"id":"1412.3587","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-11T09:54:12Z","cross_cats_sorted":[],"title_canon_sha256":"4f90484ddba1ff16d95717ae61fb3643625fd22941d51c546af61279911befac","abstract_canon_sha256":"6feebc821c78652f7c91c10b8af590eed95a653d080e13d953614fa1004ef7a6"},"schema_version":"1.0"},"canonical_sha256":"d770f52090b47588336a9442da4d226263f66ea8269679c0549ccdfb5f00cbd0","source":{"kind":"arxiv","id":"1412.3587","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.3587","created_at":"2026-05-18T02:31:30Z"},{"alias_kind":"arxiv_version","alias_value":"1412.3587v1","created_at":"2026-05-18T02:31:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.3587","created_at":"2026-05-18T02:31:30Z"},{"alias_kind":"pith_short_12","alias_value":"25YPKIEQWR2Y","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"25YPKIEQWR2YQM3K","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"25YPKIEQ","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:25YPKIEQWR2YQM3KSRBNUTJCMJ","target":"record","payload":{"canonical_record":{"source":{"id":"1412.3587","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-11T09:54:12Z","cross_cats_sorted":[],"title_canon_sha256":"4f90484ddba1ff16d95717ae61fb3643625fd22941d51c546af61279911befac","abstract_canon_sha256":"6feebc821c78652f7c91c10b8af590eed95a653d080e13d953614fa1004ef7a6"},"schema_version":"1.0"},"canonical_sha256":"d770f52090b47588336a9442da4d226263f66ea8269679c0549ccdfb5f00cbd0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:31:30.782984Z","signature_b64":"tG1mEZ6w4dbl3vMsMNEDSEUJU+iUqg71Q7zCDRJkiLDw9Qos8tWP5h29Zi3EXxiXqWyOoa39dOWBVCmTMhRqAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d770f52090b47588336a9442da4d226263f66ea8269679c0549ccdfb5f00cbd0","last_reissued_at":"2026-05-18T02:31:30.782581Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:31:30.782581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.3587","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":"2026-05-18T02:31:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7aEcMMkNuJPayXe3uy6BTcGXBVDFjjwQDeo7G05pStrls4yC+rhQQBEmNZGhlTXy8kYPLJza7hCRQNILy9A0Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:35:59.105015Z"},"content_sha256":"3f1d8aa99ce583398d792c31e2a82ee4617f281ebc4047ee45678b332ec77ff8","schema_version":"1.0","event_id":"sha256:3f1d8aa99ce583398d792c31e2a82ee4617f281ebc4047ee45678b332ec77ff8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:25YPKIEQWR2YQM3KSRBNUTJCMJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gabor systems and almost periodic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Antonio Galbis, Carmen Fern\\'andez, Paolo Boggiatto","submitted_at":"2014-12-11T09:54:12Z","abstract_excerpt":"We give a construction of Gabor type frames for suitable separable subspaces of the non-separable Hilbert spaces $AP_2({\\mathbb R})$ of almost periodic functions of one variable. Furthermore we determine a non-countable generalized frame for the whole space $AP_2({\\mathbb R}).$ We show furthermore that Bessel-type estimates hold for the $AP$ norm with respect to a countable Gabor system using suitable almost periodic norms of sequencies."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3587","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":"2026-05-18T02:31:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SqUYeYnZaR6/4tp9i537+kVXlB+VoGEtjvFGoFn2VcRmoQVD9rLDEODamWuIr+RcNOGY/xyGn3PPkhg8111TCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T07:35:59.105688Z"},"content_sha256":"5d77aed6ca66727888e51d4c147579a9fbf60bce1c36dc7aa8961a9c02e52293","schema_version":"1.0","event_id":"sha256:5d77aed6ca66727888e51d4c147579a9fbf60bce1c36dc7aa8961a9c02e52293"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ/bundle.json","state_url":"https://pith.science/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ/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":"2026-06-07T07:35:59Z","links":{"resolver":"https://pith.science/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ","bundle":"https://pith.science/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ/bundle.json","state":"https://pith.science/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/25YPKIEQWR2YQM3KSRBNUTJCMJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:25YPKIEQWR2YQM3KSRBNUTJCMJ","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":"6feebc821c78652f7c91c10b8af590eed95a653d080e13d953614fa1004ef7a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-11T09:54:12Z","title_canon_sha256":"4f90484ddba1ff16d95717ae61fb3643625fd22941d51c546af61279911befac"},"schema_version":"1.0","source":{"id":"1412.3587","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.3587","created_at":"2026-05-18T02:31:30Z"},{"alias_kind":"arxiv_version","alias_value":"1412.3587v1","created_at":"2026-05-18T02:31:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.3587","created_at":"2026-05-18T02:31:30Z"},{"alias_kind":"pith_short_12","alias_value":"25YPKIEQWR2Y","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"25YPKIEQWR2YQM3K","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"25YPKIEQ","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:5d77aed6ca66727888e51d4c147579a9fbf60bce1c36dc7aa8961a9c02e52293","target":"graph","created_at":"2026-05-18T02:31:30Z","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 give a construction of Gabor type frames for suitable separable subspaces of the non-separable Hilbert spaces $AP_2({\\mathbb R})$ of almost periodic functions of one variable. Furthermore we determine a non-countable generalized frame for the whole space $AP_2({\\mathbb R}).$ We show furthermore that Bessel-type estimates hold for the $AP$ norm with respect to a countable Gabor system using suitable almost periodic norms of sequencies.","authors_text":"Antonio Galbis, Carmen Fern\\'andez, Paolo Boggiatto","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-11T09:54:12Z","title":"Gabor systems and almost periodic functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3587","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":"sha256:3f1d8aa99ce583398d792c31e2a82ee4617f281ebc4047ee45678b332ec77ff8","target":"record","created_at":"2026-05-18T02:31:30Z","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":"6feebc821c78652f7c91c10b8af590eed95a653d080e13d953614fa1004ef7a6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-12-11T09:54:12Z","title_canon_sha256":"4f90484ddba1ff16d95717ae61fb3643625fd22941d51c546af61279911befac"},"schema_version":"1.0","source":{"id":"1412.3587","kind":"arxiv","version":1}},"canonical_sha256":"d770f52090b47588336a9442da4d226263f66ea8269679c0549ccdfb5f00cbd0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d770f52090b47588336a9442da4d226263f66ea8269679c0549ccdfb5f00cbd0","first_computed_at":"2026-05-18T02:31:30.782581Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:31:30.782581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tG1mEZ6w4dbl3vMsMNEDSEUJU+iUqg71Q7zCDRJkiLDw9Qos8tWP5h29Zi3EXxiXqWyOoa39dOWBVCmTMhRqAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:31:30.782984Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.3587","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3f1d8aa99ce583398d792c31e2a82ee4617f281ebc4047ee45678b332ec77ff8","sha256:5d77aed6ca66727888e51d4c147579a9fbf60bce1c36dc7aa8961a9c02e52293"],"state_sha256":"88d5d4454dd37829d90b6a5a26714940496ca5ba3a01e120526f7e884e47706a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bQw8zCcVW1qkaoFjNiC4eBzVG+JwaJfUfvXEQbRi1WiimrWWQjKs2WyByGZVkAUISeiYpKSH+hDdHGBbUVIoDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T07:35:59.109206Z","bundle_sha256":"592f95374ebf9d98cfc54ea7bf5a3af245c4ff344fbc23a80d5179dc376d42c0"}}