Some numerical results on best uniform rational approximation ofx ^α on [0,1]

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])} ^=1/30 _n 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.