Speed of Convergence of Certain Rational Approximations

Abstract

1. Investigating semi-discrete methods for the numerical treatment of heat-conduction problems Cody, Meinardus & Varga [1] studied the uniform approximation of e^−x by rational functions. More precisely, denoting by Π_n the collection of all real polynomials of degree at most n and by $||.|{|_{{{L_{\infty }}[0,\infty )}}}$ the supremumnorm on the positive real axis they asked for

$${\lambda _{{m,n}}}: = \mathop{{\inf }}\limits_{{\mathop{{{p_{m}} \in {\pi _{m}}}}\limits_{{{q_{m}} \in {\pi _{n}}}} }} ||{e^{{ - x}}} - \frac{{{p_{m}}(x)}}{{{q_{n}}(x)}}|{|_{{L\infty [0,\infty ]}}} $$

(1.1)

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