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Our main result shows that, for $s \\in \\mathbb{N}$, $$ E_n(f)_\\mu \\le c n^{-2s}[E_{n-2s}(\\Delta^s f)_{\\mu+2s} + E_{n}(\\Delta_0^s f)_{\\mu}], $$ where $\\Delta$ and $\\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.05515","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-09-18T17:09:11Z","cross_cats_sorted":[],"title_canon_sha256":"d6044faf6b73d92dcc618076f43ed94fcad2c4a4b49e0516114b1d92aff7317c","abstract_canon_sha256":"fd9416c10c3a80e46fd4c24723542e246fa73babb5f076518fc118a504156bb1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:15.474604Z","signature_b64":"AVQ8mHj16KW/wB8jtJ98g6ZWU/dfXyCP7VYBJOVnsqyFKoieDta26cSspNbFGiOBZPA/8cFALQ/O3AKTQm6dCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"736b152ad383bcfc5a703fa1a9db00192f2bfaece0a1259fd0ced321c0339f12","last_reissued_at":"2026-05-18T00:45:15.473903Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:15.473903Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Best polynomial approximation on the unit ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Miguel Pinar, Yuan Xu","submitted_at":"2016-09-18T17:09:11Z","abstract_excerpt":"Let $E_n(f)_\\mu$ be the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\\varpi_\\mu, \\mathbb{B}^d)$, where $\\mathbb{B}^d$ is the unit ball in $\\mathbb{R}^d$ and $\\varpi_\\mu(x) = (1-\\|x\\|^2)^\\mu$ for $\\mu > -1$. Our main result shows that, for $s \\in \\mathbb{N}$, $$ E_n(f)_\\mu \\le c n^{-2s}[E_{n-2s}(\\Delta^s f)_{\\mu+2s} + E_{n}(\\Delta_0^s f)_{\\mu}], $$ where $\\Delta$ and $\\Delta_0$ are the Laplace and Laplace-Beltrami operators, respectively. We also derive a bound when the right hand side contains odd order derivatives."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05515","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.05515","created_at":"2026-05-18T00:45:15.474015+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.05515v2","created_at":"2026-05-18T00:45:15.474015+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05515","created_at":"2026-05-18T00:45:15.474015+00:00"},{"alias_kind":"pith_short_12","alias_value":"ONVRKKWTQO6P","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"ONVRKKWTQO6PYWTQ","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"ONVRKKWT","created_at":"2026-05-18T12:30:36.002864+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE","json":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE.json","graph_json":"https://pith.science/api/pith-number/ONVRKKWTQO6PYWTQH6Q2TWYADE/graph.json","events_json":"https://pith.science/api/pith-number/ONVRKKWTQO6PYWTQH6Q2TWYADE/events.json","paper":"https://pith.science/paper/ONVRKKWT"},"agent_actions":{"view_html":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE","download_json":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE.json","view_paper":"https://pith.science/paper/ONVRKKWT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.05515&json=true","fetch_graph":"https://pith.science/api/pith-number/ONVRKKWTQO6PYWTQH6Q2TWYADE/graph.json","fetch_events":"https://pith.science/api/pith-number/ONVRKKWTQO6PYWTQH6Q2TWYADE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE/action/storage_attestation","attest_author":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE/action/author_attestation","sign_citation":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE/action/citation_signature","submit_replication":"https://pith.science/pith/ONVRKKWTQO6PYWTQH6Q2TWYADE/action/replication_record"}},"created_at":"2026-05-18T00:45:15.474015+00:00","updated_at":"2026-05-18T00:45:15.474015+00:00"}