{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:XI2BPDX7LCLFYUOVHPSYIEGCYG","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":"2eacfa93c885c2e20251243cfd28aa58ce246e01318074b4246bbfc65ceb7bb2","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2004-03-22T02:40:14Z","title_canon_sha256":"3054b620e01db7073c859a04165ef6c1730271db8063300c301ea50f626bff3a"},"schema_version":"1.0","source":{"id":"math/0403344","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0403344","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"arxiv_version","alias_value":"math/0403344v4","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0403344","created_at":"2026-05-18T01:05:26Z"},{"alias_kind":"pith_short_12","alias_value":"XI2BPDX7LCLF","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"XI2BPDX7LCLFYUOV","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"XI2BPDX7","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:2f4606f0390da18cc2ccd19d900291f511206bd4c554d650c92ca3cf74e5060c","target":"graph","created_at":"2026-05-18T01:05:26Z","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":"An inverse polynomial has a Chebyshev series expansion\n 1/\\sum(j=0..k)b_j*T_j(x)=\\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no roots in [-1,1]. If the inverse polynomial is decomposed into partial fractions, the a_n are linear combinations of simple functions of the polynomial roots. If the first k of the coefficients a_n are known, the others become linear combinations of these with expansion coefficients derived recursively from the b_j's. On a closely related theme, finding a polynomial with minimum relative error towards a given f(x) is approximately equivalent to finding the b_j in f","authors_text":"Richard J. Mathar","cross_cats":[],"headline":"","license":"","primary_cat":"math.CA","submitted_at":"2004-03-22T02:40:14Z","title":"Chebyshev Series Expansion of Inverse Polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0403344","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:ba88d2131b7797cb178867ce862e5525722174539e7df5fa1cdbb2d92c787154","target":"record","created_at":"2026-05-18T01:05:26Z","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":"2eacfa93c885c2e20251243cfd28aa58ce246e01318074b4246bbfc65ceb7bb2","cross_cats_sorted":[],"license":"","primary_cat":"math.CA","submitted_at":"2004-03-22T02:40:14Z","title_canon_sha256":"3054b620e01db7073c859a04165ef6c1730271db8063300c301ea50f626bff3a"},"schema_version":"1.0","source":{"id":"math/0403344","kind":"arxiv","version":4}},"canonical_sha256":"ba34178eff58965c51d53be58410c2c1a40e20a1fa38800f72c14f068589a106","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ba34178eff58965c51d53be58410c2c1a40e20a1fa38800f72c14f068589a106","first_computed_at":"2026-05-18T01:05:26.504146Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:26.504146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NHi0MxKurtgopy6FEg+ImnDFjtbhmFmuCpz9VTYEIgzwjPZvwDYb8hr7OulaGAIQaQYoiERQ/FpKd0NSm4LuBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:26.504752Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0403344","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba88d2131b7797cb178867ce862e5525722174539e7df5fa1cdbb2d92c787154","sha256:2f4606f0390da18cc2ccd19d900291f511206bd4c554d650c92ca3cf74e5060c"],"state_sha256":"5ecce5f156dfef7a24830c69557ad1d356f2fe24569a7f5d90517151da3e259d"}