{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:S5AH75DFKARTSZN3ZKJMSBQ6VM","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":"50e843e6e356b862281f802dd6733f93644c7845515948f55dbc6a73d0492400","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-07-09T07:00:39Z","title_canon_sha256":"7d0e71e3a9dc8db8d7fb6f4f7c69871e4cd88bb5112aa23571c969f0addb250f"},"schema_version":"1.0","source":{"id":"1907.04015","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.04015","created_at":"2026-05-17T23:41:05Z"},{"alias_kind":"arxiv_version","alias_value":"1907.04015v1","created_at":"2026-05-17T23:41:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.04015","created_at":"2026-05-17T23:41:05Z"},{"alias_kind":"pith_short_12","alias_value":"S5AH75DFKART","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_16","alias_value":"S5AH75DFKARTSZN3","created_at":"2026-05-18T12:33:27Z"},{"alias_kind":"pith_short_8","alias_value":"S5AH75DF","created_at":"2026-05-18T12:33:27Z"}],"graph_snapshots":[{"event_id":"sha256:8aa179fe3285b381029c4221d6206d91cf5a09b64b5856e1aa8eaf287304131c","target":"graph","created_at":"2026-05-17T23:41:05Z","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":"It is known that for a $\\rho$-weighted $L_q$-approximation of single variable functions $f$ with the $r$th derivatives in a $\\psi$-weighted $L_p$ space, the minimal error of approximations that use $n$ samples of $f$ is proportional to $\\|\\omega^{1/\\alpha}\\|_{L_1}^\\alpha\\|f^{(r)}\\psi\\|_{L_p}n^{-r+(1/p-1/q)_+},$ where $\\omega=\\rho/\\psi$ and $\\alpha=r-1/p+1/q.$ Moreover, the optimal sample points are determined by quantiles of $\\omega^{1/\\alpha}.$ In this paper, we show how the error of best approximations changes when the sample points are determined by a quantizer $\\kappa$ other than $\\omega.$","authors_text":"F. Pillichshammer, G.W. Wasilkowski, L. Plaskota, P. Kritzer","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-07-09T07:00:39Z","title":"On alternative quantization for doubly weighted approximation and integration over unbounded domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.04015","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":"sha256:61cfc89161d13f2718163b67dccb73c599fb4847323ea3ac5c2ea58385bf9b55","target":"record","created_at":"2026-05-17T23:41:05Z","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":"50e843e6e356b862281f802dd6733f93644c7845515948f55dbc6a73d0492400","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2019-07-09T07:00:39Z","title_canon_sha256":"7d0e71e3a9dc8db8d7fb6f4f7c69871e4cd88bb5112aa23571c969f0addb250f"},"schema_version":"1.0","source":{"id":"1907.04015","kind":"arxiv","version":1}},"canonical_sha256":"97407ff46550233965bbca92c9061eab093ed4f7e8ebed5dbd28c5877ee8eba4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97407ff46550233965bbca92c9061eab093ed4f7e8ebed5dbd28c5877ee8eba4","first_computed_at":"2026-05-17T23:41:05.961186Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:05.961186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"61TptyQn0WoKXZfUfo2WLIu22cLEd1IIaU5zsx7Kty/AVN56n6sFKstr8D2Cp72fpsyg/47qcjsEAHJX4LWVBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:05.961986Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.04015","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61cfc89161d13f2718163b67dccb73c599fb4847323ea3ac5c2ea58385bf9b55","sha256:8aa179fe3285b381029c4221d6206d91cf5a09b64b5856e1aa8eaf287304131c"],"state_sha256":"e5d90791a2aaaa8d8c95d3295c6fa8c3d2e414683183c4f7be1e52505e6db167"}