Smoothing of weights in the Bernstein approximation problem

Abstract

In 1924 S.Bernstein asked for conditions on a uniformly bounded on $\mathbb{R}$ Borel function (weight) $w:\mathbb{R}\to [0,+\infty )$ which imply the denseness of algebraic polynomials $\mathcal{P}$ in the seminormed space $C_w^0$ defined as the linear set $ \{f \in C (\mathbb{R}) \ | \ w (x) f (x) \to 0 \ \mbox{as} \ {|x| \to +\infty}\}$ equipped with the seminorm $\Vert f{\Vert }_w:={\sup }_{x\in \mathbb{R}}w(x)|f(x)|$. In 1998 A.Borichev and M.Sodin completely solved this problem for all those weights $w$ for which $\mathcal{P}$ is dense in $C_w^0$ but there exists a positive integer $n=n(w)$ such that $\mathcal{P}$ is not dense in $C_{(1+x^2{)}^{n}w}^0$. In the present paper we establish that if $\mathcal{P}$ is dense in $C_{(1+x^2{)}^{n}w}^0$ for all $n\ge 0$ then for arbitrary $\epsilon >0$ there exists a weight $W_{\epsilon }\in C^{\infty }(\mathbb{R})$ such that $\mathcal{P}$ is dense in $C_{(1+x^2{)}^n{W}_{\epsilon }}^0$ for every $n\ge 0$ and $W_{\epsilon }(x)\ge w(x)+{\mathrm{e}}^{-\epsilon |x|}$ for all $x\in \mathbb{R}$.

15 pages