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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

Best uniform rational approximation of $ x^\alpha$ on $ [0,1]$


Author: Herbert Stahl
Journal: Bull. Amer. Math. Soc. 28 (1993), 116-122
MSC: Primary 41A20
MathSciNet review: 1168517
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Abstract: A strong error estimate for the uniform rational approximation of $ {x^{\alpha}}$ on [0, 1] is given, and its proof is sketched. Let $ {E_{nn}}({x^\alpha },[0,1])$ denote the minimal approximation error in the uniform norm. Then it is shown that

$\displaystyle \mathop {\lim }\limits_{x \to \infty } {e^{2\pi \sqrt {\alpha n} }}{E_{nn}}({x^\alpha },[0,1]) = {4^{1 + \alpha }}\vert\sin \pi \alpha \vert$

holds true for each $ {\alpha > 0}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0273-0979-1993-00351-3
PII: S 0273-0979(1993)00351-3
Article copyright: © Copyright 1993 American Mathematical Society