Weighted polynomial approximation with Freud weights
The authors have shown that if \(w(x)= \exp(-| x|^ \lambda)\) and \(I_ \lambda\) be the support of an external measure associated with it then (i) for \(\lambda= 1\) for every continuous \(f\) that vanishes outside \(I_ \lambda\) there are polynomials \(P_ n\) of degree of most \(n\) such that \(w^ n P_ n\) uniformly tends to \(f\), (ii) for \(0< \lambda< 1\), a necessary and sufficient condition that a continuous function \(f\) that vanishes outside \(I_ \lambda\) be approximable by polynomials of the form \(w^ n P_ n\) is that \(f\) vanishes at the origin, (iii) for \(\lambda> 0\) if \(w^ n P_ n\) tends uniformly to \(f\) on \(I_ \lambda\) then \(f\) must vanish at the end points of \(T_ \lambda\) and actually determine the (sequence of) largest possible interval where approximation is possible. Finally, discuss approximation by weighted polynomials of the form \(W(a_ n x) P_ n(x)\).
scientific article; zbMATH DE number 672159