Rational approximation with varying weights I (Q1922537)

Let us denote by \(\delta_n(f,A) = \inf_{r \in R_n} \mid\mid 1 - (r(x)/f(x))\mid\mid_A\) the best rational relative approximation of a positive function \(f\) in the uniform norm on the real interval \(A\subset [0, +\infty)\). The authors consider the asymptotic behaviour of \(\delta_n(f_n,A)\) where \(f_n(x) = e^{-nx}\) or \(f_n(x) = x^{-n\theta}\) (\(1< \theta\)) and obtain very sharp estimates of the extreme points of the interval \(A\) when \(\delta_n \rightarrow 0\). They prove, among others, the following results: If \(\delta_n(e^{-x},[0,b_n]) \rightarrow 0\), then \(b_n\leq (1+\epsilon)2\pi n\) for any \(\epsilon >0\) and \(n\geq n(\epsilon)\) and if \(\delta_n(x^{-n\theta}, [a,1]) \rightarrow 0\) as \(n \rightarrow \infty\) and \(\theta > 1\), then \( a \geq tan^4 \left( (\pi(\theta -1))/(4\theta)\right)\). The proofs are based in complicated calculation that involve sharp results of potential theory. However, the exposition is clear and motivate several other problems.