{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MQMDLGV7TYZ7PR74HRYDVD4I2M","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":"a5908f3059af629538e6a169eb4d70a7eca62d81ed2cd1777a5b2995b9e8edab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-21T18:49:02Z","title_canon_sha256":"f7d6606d166dfa5c16a5d12783da7a82bfa0574bed11e77357b672b7a9699159"},"schema_version":"1.0","source":{"id":"1010.4554","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4554","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4554v2","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4554","created_at":"2026-05-18T03:24:42Z"},{"alias_kind":"pith_short_12","alias_value":"MQMDLGV7TYZ7","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MQMDLGV7TYZ7PR74","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MQMDLGV7","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:5b365511f44e126900e7522434c705e108a94c1bdaf64006cdcbc5a677f810e4","target":"graph","created_at":"2026-05-18T03:24:42Z","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":"Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function(RBF) approximation. The purpose of this paper is to extend what is known by deriving $L^p$ Bernstein inequalities for RBF networks on $\\mathbb{R}^d$. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding $L^p$ norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorems.","authors_text":"John Paul Ward","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-21T18:49:02Z","title":"$L^p$ Bernstein Inequalities and Inverse Theorems for RBF Approximation on $\\mathbb{R}^d$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4554","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:cfbe40a5b9221537c54282236d612737ab3bf45bd13dc7282f6ab389feea7caa","target":"record","created_at":"2026-05-18T03:24:42Z","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":"a5908f3059af629538e6a169eb4d70a7eca62d81ed2cd1777a5b2995b9e8edab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-21T18:49:02Z","title_canon_sha256":"f7d6606d166dfa5c16a5d12783da7a82bfa0574bed11e77357b672b7a9699159"},"schema_version":"1.0","source":{"id":"1010.4554","kind":"arxiv","version":2}},"canonical_sha256":"6418359abf9e33f7c7fc3c703a8f88d305b2d7d1e7d1c535712ff8481aafd216","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6418359abf9e33f7c7fc3c703a8f88d305b2d7d1e7d1c535712ff8481aafd216","first_computed_at":"2026-05-18T03:24:42.209592Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:24:42.209592Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M2Nhx81LM2ZhqTg7BhgPQqw1peq2f+RngCeR9gHwvCiT0LlB4PD9yQaLH0ljlhzOlv0uiPK99J3qGmQeso/gDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:24:42.210277Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.4554","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfbe40a5b9221537c54282236d612737ab3bf45bd13dc7282f6ab389feea7caa","sha256:5b365511f44e126900e7522434c705e108a94c1bdaf64006cdcbc5a677f810e4"],"state_sha256":"41b8c4549130850cee159b50059f69bdb78756c570bcb09ade0608a65b409586"}