Weighted polynomial approximation on the real line
One of the most important subjects in the last decades has been related to polynomial approximation on the whole real line with weight \(w(x)= \exp[-Q(x)]\), where \(Q(x)\) has polynomial growth at infinity. There are two main problems studied in the present paper: asymptotics for the Markov factors and for the rate of best approximation of \(|x|\) and Jackson-type estimates for the degree of best approximation of some classes of functions. The authors introduce the rate of best weighted approximation \[ E_n^*(f,Q)= \inf_{p\in M_n}|w(x)(f(x)- p(x))| \] in order to proceed their studies. The paper is divided into two sections. First of them, ``Asymptotics for \(M_n(Q)\) and \(E_n^* (|x|,Q)\)'' is devoted to the following result: \(M_n(Q)\sim \frac{1} {E_n^*(|x|,Q)}\sim I_n(Q)\), where \(M_n(Q)\) are so-called Markov factors and \[ I_n(Q)= \int_1^{Q^{\{-1\}}(n)} \frac{Q(t)}{t^k} dt. \] More than that, the authors gave a new proof of the upper bound for \(M_n(Q)\), which is considerably simpler than previous results and also allows to substantially relax the restrictions imposed upon the weight \(Q\). The second section, ``Order of weighted approximation on the real line'', contains the investigations of the magnitude's order of \(E_n^*(f,Q)\) in order to provide some Jackson-type estimates and to discuss lower bounds as well.
scientific article; zbMATH DE number 810857