Smoothing of weights in the Bernstein approximation problem

Abstract:

In 1924 S. Bernstein [Bull. Soc. Math. France 52 (1924), 399-410] asked for conditions on a uniformly bounded $\mathbb {R}$ Borel function (weight) $w: \mathbb {R} \to [0, +\infty )$ which imply the denseness of algebraic polynomials ${\mathcal {P} }$ in the seminormed space $C^{ 0}_{w}$ 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 $\|f\|_{w} := \sup \nolimits _{x \in {\mathbb {R}}} w(x)| f( x )|$. In 1998 A. Borichev and M. Sodin [J. Anal. Math 76 (1998), 219-264] completely solved this problem for all those weights $w$ for which ${\mathcal {P} }$ is dense in $C^{ 0}_{w}$ but for which there exists a positive integer $n=n(w)$ such that ${\mathcal {P} }$ is not dense in $C^{ 0}_{(1+x^{2})^{n} w}$. In the present paper we establish that if ${\mathcal {P} }$ is dense in $C^{ 0}_{(1+x^{2})^{n} w}$ for all $n \geq 0$, then for arbitrary $\varepsilon > 0$ there exists a weight $W_{\varepsilon } \in C^{\infty } (\mathbb {R})$ such that ${\mathcal {P}}$ is dense in $C^{ 0}_{(1+x^{2})^{n} W_{\varepsilon }}$ for every $n \geq 0$ and $W_{\varepsilon } (x) \geq w (x) + \mathrm {e}^{- \varepsilon |x|}$ for all $x\in \mathbb {R}$.