{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FNAXTKBYIPJOLU57ADAQ2BPXQD","short_pith_number":"pith:FNAXTKBY","canonical_record":{"source":{"id":"1009.4389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-09-22T15:29:00Z","cross_cats_sorted":[],"title_canon_sha256":"9c4fdafbec558aca8f33eebf4c78db0806b11ec92a3ba4d5587d38e9afc0358a","abstract_canon_sha256":"2a93ff8080ab61c374c5d199a59580f283fce08fa6caf58c3f6e76cb157ec51a"},"schema_version":"1.0"},"canonical_sha256":"2b4179a83843d2e5d3bf00c10d05f780dde8c08b95cc0b16791f67202f923340","source":{"kind":"arxiv","id":"1009.4389","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4389","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4389v1","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4389","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"pith_short_12","alias_value":"FNAXTKBYIPJO","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FNAXTKBYIPJOLU57","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FNAXTKBY","created_at":"2026-05-18T12:26:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FNAXTKBYIPJOLU57ADAQ2BPXQD","target":"record","payload":{"canonical_record":{"source":{"id":"1009.4389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-09-22T15:29:00Z","cross_cats_sorted":[],"title_canon_sha256":"9c4fdafbec558aca8f33eebf4c78db0806b11ec92a3ba4d5587d38e9afc0358a","abstract_canon_sha256":"2a93ff8080ab61c374c5d199a59580f283fce08fa6caf58c3f6e76cb157ec51a"},"schema_version":"1.0"},"canonical_sha256":"2b4179a83843d2e5d3bf00c10d05f780dde8c08b95cc0b16791f67202f923340","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:33.234389Z","signature_b64":"wfewSuJ6JBXCP+ToUeCSMn3xlYpZoIMGW0oUvqI4Pt3bP08UDVQxPyujzCrxYiyNDjn1E24zvPXJ94hOADLyAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b4179a83843d2e5d3bf00c10d05f780dde8c08b95cc0b16791f67202f923340","last_reissued_at":"2026-05-18T04:40:33.233847Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:33.233847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.4389","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":"2026-05-18T04:40:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vqtIXqoq11i1KjYgjJKmARp1QOPhCkwskrBmCCqvTBdVhdjAvIaKUbBzrGpG6Q8oRR3Xec/mfS21P6xtcj5GDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T19:24:44.041525Z"},"content_sha256":"4ad34ecce358d01354745caa04cc13600b83a164396ed2c7c96190bf58c6526f","schema_version":"1.0","event_id":"sha256:4ad34ecce358d01354745caa04cc13600b83a164396ed2c7c96190bf58c6526f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FNAXTKBYIPJOLU57ADAQ2BPXQD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dinh D\\~ung","submitted_at":"2010-09-22T15:29:00Z","abstract_excerpt":"Let $\\xi = \\{x^j\\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube ${\\II}^d:=[0,1]^d$, and $\\Phi = \\{\\phi_j\\}_{j =1}^n$ a family of $n$ functions on ${\\II}^d$. We define the linear sampling algorithm $L_n(\\Phi,\\xi,\\cdot)$ for an approximate recovery of a continuous function $f$ on ${\\II}^d$ from the sampled values $f(x^1), ..., f(x^n)$, by $$L_n(\\Phi,\\xi,f)\\ := \\ \\sum_{j=1}^n f(x^j)\\phi_j$$.\n  For the Besov class $B^\\alpha_{p,\\theta}$ of mixed smoothness $\\alpha$ (defined as the unit ball of the Besov space $\\MB$), to study optimality of $L_n(\\Phi,\\xi,\\cdot)$ in $L_q({\\II}^d)$ we use the quan"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4389","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":"2026-05-18T04:40:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c+0WOpVlYwvvpVtp6LQVfEqLM1H060pLOLqVxXmw+5G+c7ek11bWxb0VXiSkHpmgir2N8AweyaHbNonTqmpIAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T19:24:44.041877Z"},"content_sha256":"48d35c337d2e22d44ca906c1f70a268b1b93a3e162d61749eebdfff51f454ac2","schema_version":"1.0","event_id":"sha256:48d35c337d2e22d44ca906c1f70a268b1b93a3e162d61749eebdfff51f454ac2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD/bundle.json","state_url":"https://pith.science/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD/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":"2026-06-02T19:24:44Z","links":{"resolver":"https://pith.science/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD","bundle":"https://pith.science/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD/bundle.json","state":"https://pith.science/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FNAXTKBYIPJOLU57ADAQ2BPXQD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FNAXTKBYIPJOLU57ADAQ2BPXQD","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":"2a93ff8080ab61c374c5d199a59580f283fce08fa6caf58c3f6e76cb157ec51a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-09-22T15:29:00Z","title_canon_sha256":"9c4fdafbec558aca8f33eebf4c78db0806b11ec92a3ba4d5587d38e9afc0358a"},"schema_version":"1.0","source":{"id":"1009.4389","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.4389","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"arxiv_version","alias_value":"1009.4389v1","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.4389","created_at":"2026-05-18T04:40:33Z"},{"alias_kind":"pith_short_12","alias_value":"FNAXTKBYIPJO","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_16","alias_value":"FNAXTKBYIPJOLU57","created_at":"2026-05-18T12:26:07Z"},{"alias_kind":"pith_short_8","alias_value":"FNAXTKBY","created_at":"2026-05-18T12:26:07Z"}],"graph_snapshots":[{"event_id":"sha256:48d35c337d2e22d44ca906c1f70a268b1b93a3e162d61749eebdfff51f454ac2","target":"graph","created_at":"2026-05-18T04:40:33Z","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":"Let $\\xi = \\{x^j\\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube ${\\II}^d:=[0,1]^d$, and $\\Phi = \\{\\phi_j\\}_{j =1}^n$ a family of $n$ functions on ${\\II}^d$. We define the linear sampling algorithm $L_n(\\Phi,\\xi,\\cdot)$ for an approximate recovery of a continuous function $f$ on ${\\II}^d$ from the sampled values $f(x^1), ..., f(x^n)$, by $$L_n(\\Phi,\\xi,f)\\ := \\ \\sum_{j=1}^n f(x^j)\\phi_j$$.\n  For the Besov class $B^\\alpha_{p,\\theta}$ of mixed smoothness $\\alpha$ (defined as the unit ball of the Besov space $\\MB$), to study optimality of $L_n(\\Phi,\\xi,\\cdot)$ in $L_q({\\II}^d)$ we use the quan","authors_text":"Dinh D\\~ung","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-09-22T15:29:00Z","title":"B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4389","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":"sha256:4ad34ecce358d01354745caa04cc13600b83a164396ed2c7c96190bf58c6526f","target":"record","created_at":"2026-05-18T04:40:33Z","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":"2a93ff8080ab61c374c5d199a59580f283fce08fa6caf58c3f6e76cb157ec51a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-09-22T15:29:00Z","title_canon_sha256":"9c4fdafbec558aca8f33eebf4c78db0806b11ec92a3ba4d5587d38e9afc0358a"},"schema_version":"1.0","source":{"id":"1009.4389","kind":"arxiv","version":1}},"canonical_sha256":"2b4179a83843d2e5d3bf00c10d05f780dde8c08b95cc0b16791f67202f923340","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2b4179a83843d2e5d3bf00c10d05f780dde8c08b95cc0b16791f67202f923340","first_computed_at":"2026-05-18T04:40:33.233847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:40:33.233847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wfewSuJ6JBXCP+ToUeCSMn3xlYpZoIMGW0oUvqI4Pt3bP08UDVQxPyujzCrxYiyNDjn1E24zvPXJ94hOADLyAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:40:33.234389Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.4389","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4ad34ecce358d01354745caa04cc13600b83a164396ed2c7c96190bf58c6526f","sha256:48d35c337d2e22d44ca906c1f70a268b1b93a3e162d61749eebdfff51f454ac2"],"state_sha256":"bba1d7eb8e7ad844a786f4fb4af8fe692fb012891277808f74c06350813d5767"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RVNW3zmO70RdkkUGkhl1rHLBUCW9S6iG0e/0yC2HebsDDBk+1B682bM4uFyPQsLPxZQD6V/7bkCbLMny3zQQDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T19:24:44.043904Z","bundle_sha256":"4c9fe737404faa5a0d9b39f20a7f92a16cd4a1a4c593988c8e66fb9587f6016b"}}