{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XMARR7PQQRL5QQ7MCOJ4E2BUZJ","short_pith_number":"pith:XMARR7PQ","canonical_record":{"source":{"id":"1110.1057","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-05T17:53:49Z","cross_cats_sorted":[],"title_canon_sha256":"0e6d78a30057353b80474ef1ed5fe0eefa8880ceec8d78f9967142b92aea174c","abstract_canon_sha256":"d1e431663dad99d6d0d896f322a137db72ff776248a4e58fc683b7e641c9d8fb"},"schema_version":"1.0"},"canonical_sha256":"bb0118fdf08457d843ec1393c26834ca4928e7639d670ba8422140f6226c3ba5","source":{"kind":"arxiv","id":"1110.1057","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1057","created_at":"2026-05-18T03:58:51Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1057v2","created_at":"2026-05-18T03:58:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1057","created_at":"2026-05-18T03:58:51Z"},{"alias_kind":"pith_short_12","alias_value":"XMARR7PQQRL5","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XMARR7PQQRL5QQ7M","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XMARR7PQ","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XMARR7PQQRL5QQ7MCOJ4E2BUZJ","target":"record","payload":{"canonical_record":{"source":{"id":"1110.1057","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-05T17:53:49Z","cross_cats_sorted":[],"title_canon_sha256":"0e6d78a30057353b80474ef1ed5fe0eefa8880ceec8d78f9967142b92aea174c","abstract_canon_sha256":"d1e431663dad99d6d0d896f322a137db72ff776248a4e58fc683b7e641c9d8fb"},"schema_version":"1.0"},"canonical_sha256":"bb0118fdf08457d843ec1393c26834ca4928e7639d670ba8422140f6226c3ba5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:58:51.194026Z","signature_b64":"cWpOreE4d6PcqvDETUfeRo02/y6A0Bre+SIQg2PL+2/QX09V87HUC7qDrgEzBCRTxoE0i03TAnFS07yxIjRBAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb0118fdf08457d843ec1393c26834ca4928e7639d670ba8422140f6226c3ba5","last_reissued_at":"2026-05-18T03:58:51.193329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:58:51.193329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.1057","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:58:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ywFTNf1fhsnJeLnqibH1XaiBKJ0Wf1aiqcNd1w7fR3wvw2cW52ngy9IDyKRDpY9MolSQbUrtW4BgEet7cDgDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T11:39:49.868294Z"},"content_sha256":"f4f06fd62babd78c8afe3b2a4299b14bc342f54c3fc74002f23fe93c16b8d16d","schema_version":"1.0","event_id":"sha256:f4f06fd62babd78c8afe3b2a4299b14bc342f54c3fc74002f23fe93c16b8d16d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XMARR7PQQRL5QQ7MCOJ4E2BUZJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Continuous and Discrete Fourier Frames for Fractal Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Deguang Han, Dorin Ervin Dutkay, Eric Weber","submitted_at":"2011-10-05T17:53:49Z","abstract_excerpt":"Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1057","kind":"arxiv","version":2},"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":"2026-05-18T03:58:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z2ZVqF205G7Wsg7TfRhyTY50/jXAggBs9A3rxkOOPBQprXm7siUaTFdW6iQV3GqjUIOCgJTKaZMDtzhnRns+Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T11:39:49.868660Z"},"content_sha256":"c3f48a1f4c3a7c2ae1ee76c7c0a6d946d870b1de54cd1ae5d90cb6d8d55cc6b5","schema_version":"1.0","event_id":"sha256:c3f48a1f4c3a7c2ae1ee76c7c0a6d946d870b1de54cd1ae5d90cb6d8d55cc6b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ/bundle.json","state_url":"https://pith.science/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ/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":"2026-05-20T11:39:49Z","links":{"resolver":"https://pith.science/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ","bundle":"https://pith.science/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ/bundle.json","state":"https://pith.science/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XMARR7PQQRL5QQ7MCOJ4E2BUZJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XMARR7PQQRL5QQ7MCOJ4E2BUZJ","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":"d1e431663dad99d6d0d896f322a137db72ff776248a4e58fc683b7e641c9d8fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-05T17:53:49Z","title_canon_sha256":"0e6d78a30057353b80474ef1ed5fe0eefa8880ceec8d78f9967142b92aea174c"},"schema_version":"1.0","source":{"id":"1110.1057","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1057","created_at":"2026-05-18T03:58:51Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1057v2","created_at":"2026-05-18T03:58:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1057","created_at":"2026-05-18T03:58:51Z"},{"alias_kind":"pith_short_12","alias_value":"XMARR7PQQRL5","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"XMARR7PQQRL5QQ7M","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"XMARR7PQ","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:c3f48a1f4c3a7c2ae1ee76c7c0a6d946d870b1de54cd1ae5d90cb6d8d55cc6b5","target":"graph","created_at":"2026-05-18T03:58:51Z","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":"Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel ","authors_text":"Deguang Han, Dorin Ervin Dutkay, Eric Weber","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-05T17:53:49Z","title":"Continuous and Discrete Fourier Frames for Fractal Measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1057","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:f4f06fd62babd78c8afe3b2a4299b14bc342f54c3fc74002f23fe93c16b8d16d","target":"record","created_at":"2026-05-18T03:58:51Z","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":"d1e431663dad99d6d0d896f322a137db72ff776248a4e58fc683b7e641c9d8fb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-10-05T17:53:49Z","title_canon_sha256":"0e6d78a30057353b80474ef1ed5fe0eefa8880ceec8d78f9967142b92aea174c"},"schema_version":"1.0","source":{"id":"1110.1057","kind":"arxiv","version":2}},"canonical_sha256":"bb0118fdf08457d843ec1393c26834ca4928e7639d670ba8422140f6226c3ba5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb0118fdf08457d843ec1393c26834ca4928e7639d670ba8422140f6226c3ba5","first_computed_at":"2026-05-18T03:58:51.193329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:58:51.193329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cWpOreE4d6PcqvDETUfeRo02/y6A0Bre+SIQg2PL+2/QX09V87HUC7qDrgEzBCRTxoE0i03TAnFS07yxIjRBAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:58:51.194026Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1057","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f4f06fd62babd78c8afe3b2a4299b14bc342f54c3fc74002f23fe93c16b8d16d","sha256:c3f48a1f4c3a7c2ae1ee76c7c0a6d946d870b1de54cd1ae5d90cb6d8d55cc6b5"],"state_sha256":"a4b2433566eb466ae705e1e0568eeae08ab7ad38a345fa5cb8a85a2f9296c532"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kaFPTpyEsq5cOPLBIseSSlZcKTxSIo2JkmhwBFoAHps0aMRL+EaQ1jmLKn9nO9/jeA8ZocY39zBLAlC5sf/ACw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T11:39:49.870569Z","bundle_sha256":"c33db31c64d2b216ce40457589212c29ca1afb244c859fc5953c1ccc560f4b66"}}