{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:RBJE6JIWGDXVTQGPDOLKMVJ2J3","short_pith_number":"pith:RBJE6JIW","canonical_record":{"source":{"id":"1512.05923","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-18T12:27:30Z","cross_cats_sorted":[],"title_canon_sha256":"18286628f61dae0c3f832f587d5c48baef22f14fb715d31324c5e08ecabf1d15","abstract_canon_sha256":"b2187adcaccc05ec64350e22c073bd652c333eb8c8068cd101be44be86134671"},"schema_version":"1.0"},"canonical_sha256":"88524f251630ef59c0cf1b96a6553a4ee84eb188abbb190510c0ab220fff34d9","source":{"kind":"arxiv","id":"1512.05923","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05923","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05923v1","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05923","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"pith_short_12","alias_value":"RBJE6JIWGDXV","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RBJE6JIWGDXVTQGP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RBJE6JIW","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:RBJE6JIWGDXVTQGPDOLKMVJ2J3","target":"record","payload":{"canonical_record":{"source":{"id":"1512.05923","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-18T12:27:30Z","cross_cats_sorted":[],"title_canon_sha256":"18286628f61dae0c3f832f587d5c48baef22f14fb715d31324c5e08ecabf1d15","abstract_canon_sha256":"b2187adcaccc05ec64350e22c073bd652c333eb8c8068cd101be44be86134671"},"schema_version":"1.0"},"canonical_sha256":"88524f251630ef59c0cf1b96a6553a4ee84eb188abbb190510c0ab220fff34d9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:42.723643Z","signature_b64":"8Wk7wRuRsQLBvXh2zQqux5d/hdnmFllAiT+OhyVDBpmntB0YbfNEnCX9UKbsbgz3oBbmMBVGde7xPRvOsw/3Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"88524f251630ef59c0cf1b96a6553a4ee84eb188abbb190510c0ab220fff34d9","last_reissued_at":"2026-05-18T01:01:42.723029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:42.723029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.05923","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":"2026-05-18T01:01:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aCl2F/0aB1Dg05+0yjqKeWYHj87/IBAwIu/VCdtMzOpBdySCSVRvmg7llbchPonVkMtDi5PX0pJg5+Y1Yi20Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:44:25.638501Z"},"content_sha256":"362633a6fadd29e789ba8d172ef7e5ea8773e4d14b876747dd4b7f3882fb463b","schema_version":"1.0","event_id":"sha256:362633a6fadd29e789ba8d172ef7e5ea8773e4d14b876747dd4b7f3882fb463b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:RBJE6JIWGDXVTQGPDOLKMVJ2J3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Operator Reproducing Kernel Hilbert Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Rui Wang, Yuesheng Xu","submitted_at":"2015-12-18T12:27:30Z","abstract_excerpt":"Motivated by the need of processing functional-valued data, or more general, operatorvalued data, we introduce the notion of the operator reproducing kernel Hilbert space (ORKHS). This space admits a unique operator reproducing kernel which reproduces a family of continuous linear operators on the space. The theory of ORKHSs and the associated operator reproducing kernels are established. A special class of ORKHSs, known as the perfect ORKHSs, are studied, which reproduce the family of the standard point-evaluation operators and at the same time another different family of continuous linear op"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05923","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":"2026-05-18T01:01:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xg1fHjFTYEOyhwq6ABbHSpOxyZG00BA2jMxx73uBJeIPpt4P8L3hQROXp75vfB7K1VD7QhNEywwqlHiB7qgECw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:44:25.638855Z"},"content_sha256":"9440672086659c40dd51d8af430f94324dd4dc343e74f7bb37b343ca3a5462e0","schema_version":"1.0","event_id":"sha256:9440672086659c40dd51d8af430f94324dd4dc343e74f7bb37b343ca3a5462e0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3/bundle.json","state_url":"https://pith.science/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3/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":"2026-06-02T15:44:25Z","links":{"resolver":"https://pith.science/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3","bundle":"https://pith.science/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3/bundle.json","state":"https://pith.science/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RBJE6JIWGDXVTQGPDOLKMVJ2J3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RBJE6JIWGDXVTQGPDOLKMVJ2J3","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":"b2187adcaccc05ec64350e22c073bd652c333eb8c8068cd101be44be86134671","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-18T12:27:30Z","title_canon_sha256":"18286628f61dae0c3f832f587d5c48baef22f14fb715d31324c5e08ecabf1d15"},"schema_version":"1.0","source":{"id":"1512.05923","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.05923","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"arxiv_version","alias_value":"1512.05923v1","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05923","created_at":"2026-05-18T01:01:42Z"},{"alias_kind":"pith_short_12","alias_value":"RBJE6JIWGDXV","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RBJE6JIWGDXVTQGP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RBJE6JIW","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:9440672086659c40dd51d8af430f94324dd4dc343e74f7bb37b343ca3a5462e0","target":"graph","created_at":"2026-05-18T01:01:42Z","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 need of processing functional-valued data, or more general, operatorvalued data, we introduce the notion of the operator reproducing kernel Hilbert space (ORKHS). This space admits a unique operator reproducing kernel which reproduces a family of continuous linear operators on the space. The theory of ORKHSs and the associated operator reproducing kernels are established. A special class of ORKHSs, known as the perfect ORKHSs, are studied, which reproduce the family of the standard point-evaluation operators and at the same time another different family of continuous linear op","authors_text":"Rui Wang, Yuesheng Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-18T12:27:30Z","title":"Operator Reproducing Kernel Hilbert Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05923","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":"sha256:362633a6fadd29e789ba8d172ef7e5ea8773e4d14b876747dd4b7f3882fb463b","target":"record","created_at":"2026-05-18T01:01:42Z","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":"b2187adcaccc05ec64350e22c073bd652c333eb8c8068cd101be44be86134671","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-12-18T12:27:30Z","title_canon_sha256":"18286628f61dae0c3f832f587d5c48baef22f14fb715d31324c5e08ecabf1d15"},"schema_version":"1.0","source":{"id":"1512.05923","kind":"arxiv","version":1}},"canonical_sha256":"88524f251630ef59c0cf1b96a6553a4ee84eb188abbb190510c0ab220fff34d9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"88524f251630ef59c0cf1b96a6553a4ee84eb188abbb190510c0ab220fff34d9","first_computed_at":"2026-05-18T01:01:42.723029Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:42.723029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Wk7wRuRsQLBvXh2zQqux5d/hdnmFllAiT+OhyVDBpmntB0YbfNEnCX9UKbsbgz3oBbmMBVGde7xPRvOsw/3Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:42.723643Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.05923","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:362633a6fadd29e789ba8d172ef7e5ea8773e4d14b876747dd4b7f3882fb463b","sha256:9440672086659c40dd51d8af430f94324dd4dc343e74f7bb37b343ca3a5462e0"],"state_sha256":"0287efb5090d7ec19b94b2f1e36d53eb7ba06aa13b4de163b6f482d346e02315"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m+BIfoIjycI1SfXFoYE8nNNHKUPJ9zxHMibjMA7LLp/zdDGghMwf6KLPGkYvxI6gkU50+9V1xLWmoZ/qapUiAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T15:44:25.640946Z","bundle_sha256":"2fe4f1dc14eb7856cc4a07800924444daf935d05cef6d35543e8b34aebfc1f89"}}