ISSN:
1572-9265
Keywords:
AMS (MOS)
;
41A20
;
41A50
;
Rational approximation
;
best approximation
;
Remez algorithm
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract Let α be a positive number, and letE n,n (x α;[0,1]) denote the error of best uniform rational approximation from π n,n tox α on the interval [0,1]. We rigorously determined the numbers {E n,n (x α;[0,1])} n =1/30 for six values of α in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of α, Richardson's extrapolation was applied to the products $$\{ e^{\pi \sqrt {4\alpha n} } E_{n,n} (x^\alpha ;[0,1])\} _{n = 1}^{30} $$ to obtain estimates of $$\lambda (\alpha ): = \mathop {\lim }\limits_{n \to \infty } e^{\pi \sqrt {4\alpha n} } E_{n,n} (x^\alpha ;[0,1]) (\alpha 〉 0).$$ These estimates give rise to two interesting new conjectures in the theory of rational approximation.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02145384
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