Rational approximation with varying weights. II
This paper is a continuation of the first part published in [\textit{P. Borwein, E. A. Rakhmanov} and \textit{E. B. Saff}, Constructive Approximation 12, No. 2, 223-240 (1996; Zbl 0870.41010)]. Given \(\alpha >0\), \(\beta \geq 0\) and the varying weight \(\omega^n(x):= e^{nx}\), the authors prove that any real-valued function \(f \in C[0,a]\) is the uniform limit of some sequence \((\omega^nr_n)\), where \(r_n = p_n/q_n\) is a rational function with real coefficients, \(\text{deg } p_n \leq [\alpha n]\), \(\text{deg } q_n \leq [\beta n]\) and \(a\) satisfies a certain upper bound that depends on \(\alpha\) and \(\beta\). Indeed, approximation is not longer possible if \(a\) violates the upper bound requirement. The approximation by sequences of the form \((\omega^n /q_n)\) corresponds to \(\alpha =0\) and \(\beta >0\). The authors prove that in this case a real-valued function \(f \in C[0,a]\) is uniformly approximable if and only if \(f\) does not change sign on \([0,a]\). Similar results are given for rational approximation with varying weight of the form \(\omega^n(x)=x^{\theta n}\), \(\theta >1\) and \(f \in C[b,1]\), where \(b\) satisfies a certain lower bound that depends on \(\alpha\) and \(\beta\).
scientific article; zbMATH DE number 1146515