{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XG3GCZGOIJBEDHFHIORKVPLRXN","short_pith_number":"pith:XG3GCZGO","canonical_record":{"source":{"id":"1603.01937","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-03-07T05:05:10Z","cross_cats_sorted":[],"title_canon_sha256":"d54e615cd6c22d06c16f23981fac84f71d5dd6a89f816cfb009fe3715109dbe3","abstract_canon_sha256":"c0c845a1810a2013a58d070d94a91bac0958e183ec556c2a46d1e03a2b2a7bc4"},"schema_version":"1.0"},"canonical_sha256":"b9b66164ce4242419ca743a2aabd71bb61c2df654772be5672b9e16ecc4b2ba2","source":{"kind":"arxiv","id":"1603.01937","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.01937","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"arxiv_version","alias_value":"1603.01937v4","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.01937","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"pith_short_12","alias_value":"XG3GCZGOIJBE","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XG3GCZGOIJBEDHFH","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XG3GCZGO","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XG3GCZGOIJBEDHFHIORKVPLRXN","target":"record","payload":{"canonical_record":{"source":{"id":"1603.01937","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-03-07T05:05:10Z","cross_cats_sorted":[],"title_canon_sha256":"d54e615cd6c22d06c16f23981fac84f71d5dd6a89f816cfb009fe3715109dbe3","abstract_canon_sha256":"c0c845a1810a2013a58d070d94a91bac0958e183ec556c2a46d1e03a2b2a7bc4"},"schema_version":"1.0"},"canonical_sha256":"b9b66164ce4242419ca743a2aabd71bb61c2df654772be5672b9e16ecc4b2ba2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:31.252811Z","signature_b64":"YZD54CglXnpQgtzCJkQyvNx1aPyl4RSkD4I+LIFvVv3WFeTJLUD6H18ZR/ra5TOQpQx6TXjK8aP3phEfUgbfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b9b66164ce4242419ca743a2aabd71bb61c2df654772be5672b9e16ecc4b2ba2","last_reissued_at":"2026-05-18T00:56:31.252050Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:31.252050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.01937","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:56:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"APqXhVL+fOKYdY9ZR4h/0VoCkvqZzVq3ZM5dKYS3KuTitxSDpMCh5zV4hdZaJgXZXh5gsaJK2tdXOVe4fKMUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:48:56.080590Z"},"content_sha256":"8a2d5577b1ef0ff3604ef88de197e065efb9869015602607f3efbdcf6273d37d","schema_version":"1.0","event_id":"sha256:8a2d5577b1ef0ff3604ef88de197e065efb9869015602607f3efbdcf6273d37d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XG3GCZGOIJBEDHFHIORKVPLRXN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"B-spline quasi-interpolation sampling representation and sampling recovery in Sobolev spaces of mixed smoothness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Dinh D\\~ung","submitted_at":"2016-03-07T05:05:10Z","abstract_excerpt":"We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces $W^r_p$ of mixed smoothness $r$, established estimates of the approximation error of recovery in $L_q$-norm of functions from the unit ball $U^r_p$ in the spaces $W^r_p$ by linear sampling algorithms based on this representation, the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width $r^s_n(U^r_p, L_q)$ and sampling width $r_n(U^r_p, L_q)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01937","kind":"arxiv","version":4},"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":"2026-05-18T00:56:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J/maWcH11O+7hwRbzUNNDXg1zNmogsndWLR0yf8yDPpMOyTBMC7dslxWpG5pImXXHY6JVo2d4Eqx3TCPgzxrCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:48:56.080934Z"},"content_sha256":"d85c3bd0db78e5131cd6fbab7a679b564e5503228873b34ad60f8a42fdd133ec","schema_version":"1.0","event_id":"sha256:d85c3bd0db78e5131cd6fbab7a679b564e5503228873b34ad60f8a42fdd133ec"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XG3GCZGOIJBEDHFHIORKVPLRXN/bundle.json","state_url":"https://pith.science/pith/XG3GCZGOIJBEDHFHIORKVPLRXN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XG3GCZGOIJBEDHFHIORKVPLRXN/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":"2026-06-02T20:48:56Z","links":{"resolver":"https://pith.science/pith/XG3GCZGOIJBEDHFHIORKVPLRXN","bundle":"https://pith.science/pith/XG3GCZGOIJBEDHFHIORKVPLRXN/bundle.json","state":"https://pith.science/pith/XG3GCZGOIJBEDHFHIORKVPLRXN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XG3GCZGOIJBEDHFHIORKVPLRXN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XG3GCZGOIJBEDHFHIORKVPLRXN","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":"c0c845a1810a2013a58d070d94a91bac0958e183ec556c2a46d1e03a2b2a7bc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-03-07T05:05:10Z","title_canon_sha256":"d54e615cd6c22d06c16f23981fac84f71d5dd6a89f816cfb009fe3715109dbe3"},"schema_version":"1.0","source":{"id":"1603.01937","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.01937","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"arxiv_version","alias_value":"1603.01937v4","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.01937","created_at":"2026-05-18T00:56:31Z"},{"alias_kind":"pith_short_12","alias_value":"XG3GCZGOIJBE","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XG3GCZGOIJBEDHFH","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XG3GCZGO","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:d85c3bd0db78e5131cd6fbab7a679b564e5503228873b34ad60f8a42fdd133ec","target":"graph","created_at":"2026-05-18T00:56:31Z","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 proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces $W^r_p$ of mixed smoothness $r$, established estimates of the approximation error of recovery in $L_q$-norm of functions from the unit ball $U^r_p$ in the spaces $W^r_p$ by linear sampling algorithms based on this representation, the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width $r^s_n(U^r_p, L_q)$ and sampling width $r_n(U^r_p, L_q)$.","authors_text":"Dinh D\\~ung","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-03-07T05:05:10Z","title":"B-spline quasi-interpolation sampling representation and sampling recovery in Sobolev spaces of mixed smoothness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01937","kind":"arxiv","version":4},"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":"sha256:8a2d5577b1ef0ff3604ef88de197e065efb9869015602607f3efbdcf6273d37d","target":"record","created_at":"2026-05-18T00:56:31Z","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":"c0c845a1810a2013a58d070d94a91bac0958e183ec556c2a46d1e03a2b2a7bc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2016-03-07T05:05:10Z","title_canon_sha256":"d54e615cd6c22d06c16f23981fac84f71d5dd6a89f816cfb009fe3715109dbe3"},"schema_version":"1.0","source":{"id":"1603.01937","kind":"arxiv","version":4}},"canonical_sha256":"b9b66164ce4242419ca743a2aabd71bb61c2df654772be5672b9e16ecc4b2ba2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b9b66164ce4242419ca743a2aabd71bb61c2df654772be5672b9e16ecc4b2ba2","first_computed_at":"2026-05-18T00:56:31.252050Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:31.252050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YZD54CglXnpQgtzCJkQyvNx1aPyl4RSkD4I+LIFvVv3WFeTJLUD6H18ZR/ra5TOQpQx6TXjK8aP3phEfUgbfDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:31.252811Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.01937","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a2d5577b1ef0ff3604ef88de197e065efb9869015602607f3efbdcf6273d37d","sha256:d85c3bd0db78e5131cd6fbab7a679b564e5503228873b34ad60f8a42fdd133ec"],"state_sha256":"274c24b303865ab76023a8fa4ec0f4aa2e4fd02bd75c85f12ce798845db07146"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3n0tzXiARXpogUGjLS0iQyWix1+psv0OlRHWL07A69zO29+h8OJTrb9HbsT1hFn9ups5/AV1w8TvOHS6hFLABg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T20:48:56.083124Z","bundle_sha256":"a206c44c80b85e13fdd4593c20f19c01172e27c38a4ce7c6823799eb28ddca49"}}