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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Behavior of polynomials of best uniform approximation


Authors: E. B. Saff and V. Totik
Journal: Trans. Amer. Math. Soc. 316 (1989), 567-593
MSC: Primary 30E10; Secondary 41A25
MathSciNet review: 961628
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Abstract: We investigate the asymptotic behavior of the polynomials $ \{ {P_n}(f)\} _0^\infty $ of best uniform approximation to a function $ f$ that is continuous on a compact set $ K$ of the complex plane $ {\mathbf{C}}$ and analytic in the interior of $ K$, where $ K$ has connected complement. For example, we show that for "most" functions $ f$, the error $ f - {P_n}(f)$ does not decrease faster at interior points of $ K$ than on $ K$ itself. We also describe the possible limit functions for the normalized error $ (f - {P_n}(f))/{E_n}$, where $ {E_n}: = \vert\vert f - {P_n}(f)\vert{\vert _K}$, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to $ f$ on $ K$ exist that converge more rapidly at the interior points of $ K$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0961628-3
PII: S 0002-9947(1989)0961628-3
Article copyright: © Copyright 1989 American Mathematical Society