# Rational approximation with varying weights I

Borwein, Peter and Rakhmanov, E.A. and Saff, E.B. (1995) Rational approximation with varying weights I. [Preprint]

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We investigate two problems concerning uniform approximation by weighted rationals $\{w^n r_n\}_{n=1}^\infty$, where $r_n=p_n/q_n$ is a rational function of order $n$. Namely, for $w(x):=e^x$ we prove that uniform convergence to 1 of $w^n r_n$ is not possible on any interval $[0,a]$ with $a > 2 \pi$. For $w(x):=x^\theta,\; \theta > 1$, we show that uniform convergence to 1 of $w^n r_n$ is not possible on any interval $[b,1]$ with $b< \tan^4 (\pi (\theta -1)/4 \theta)$. (The latter result can be interpreted as a rational analogue of results concerning incomplete polynomials".) More generally, for $\alpha \geq 0,\, \beta \geq 0,\, \alpha + \beta > 0$, we investigate for $w(x)=e^x$ and $w(x)=x^\theta$, the possibility of approximation by $\{w^n p_n/q_n\}_{n=1}^\infty$, where ${\rm deg}\, p_n \leq \alpha n, {\rm deg}\, q_n \leq \beta n$. The analysis utilizes potential theoretic methods. These are essentially sharp results though this will not be established in this paper.