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Our main result gives a sharp estimate of $E_n(f)_{\\alpha,\\beta,\\gamma}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. The result also leads to a characterization of $E_n(f)_{\\alpha,\\beta,\\gamma}$ by a weighted $K"},"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":"1711.04756","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-11-13T18:49:19Z","cross_cats_sorted":[],"title_canon_sha256":"a4c03f2747c4764ff991e79bbc701b36af89495d4cbaad91c21c2e4bde9d5bc6","abstract_canon_sha256":"7742029426bbfcab5057440cec5a80ceb8e0f0097d02c0e2bb29af059bbc9c89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:03.680923Z","signature_b64":"hTbfGV2TakF8smCDkvPV3qoahD7dLMoA+ErLmUGlImFidQaAvT1FrIemua35W5wPM1tpe+KzP815vtr1fkx/Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb15f25a932b50a404fd149736925f191d712670e54f45910edc3429123d2bc5","last_reissued_at":"2026-05-17T23:55:03.680391Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:03.680391Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Best polynomial approximation on the triangle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Christian Krattenthaler, Han Feng, Yuan Xu","submitted_at":"2017-11-13T18:49:19Z","abstract_excerpt":"Let $E_n(f)_{\\alpha,\\beta,\\gamma}$ denote the error of best approximation by polynomials of degree at most $n$ in the space $L^2(\\varpi_{\\alpha,\\beta,\\gamma})$ on the triangle $\\{(x,y): x, y \\ge 0, x+y \\le 1\\}$, where $\\varpi_{\\alpha,\\beta,\\gamma}(x,y) := x^\\alpha y ^\\beta (1-x-y)^\\gamma$ for $\\alpha,\\beta,\\gamma > -1$. Our main result gives a sharp estimate of $E_n(f)_{\\alpha,\\beta,\\gamma}$ in terms of the error of best approximation for higher order derivatives of $f$ in appropriate Sobolev spaces. 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