De Branges' theorem on approximation problems of Bernstein type (Q2852320)

Let \(W: {\mathbb R}\mapsto (0, +\infty)\) be a continuous function, and assume that \(\lim_{|x|\to\infty} {x^n}/{W(x)}=0\) for any \(n\in {\mathbb N}\). Let \(C_0(W)\) be the space of all continuous functions on \(\mathbb R\) such that \(\lim_{|x|\to\infty}\frac{|f(x)|}{W(x)}=0\) (here \(\frac{f(x)}{W(x)}\) is understood as zero if \(W(x)=\infty\)) endowed with the semi-norm \(\|f\|_{C_0(W)}:=\sup_{x\in {\mathbb R}} {|f(x)|}/{W(x)}\). The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in the space \(C_0(W)\). A theorem of de Branges characterizes non-density by the existence of an entire function of Krein class being related with the weight in a certain way. In the paper under review, the authors consider the approximation in weighted \(C_0\)-spaces by functions belonging to a prescribed subspace of entire functions which is solely assumed to be invariant under the division of zeros and passing from \(F(z)\) to \(\overline{F(z)}\), and establish the precise analogue of de Branges' theorems. For the proof they follow the lines of de Branges' original proof and employ some results of Pitt.