Smoothing of weights in the Bernstein approximation problem

TL;DR

This paper investigates conditions under which algebraic polynomials are dense in weighted function spaces, extending previous results by constructing smooth weights that preserve density across all polynomial degrees.

Contribution

It establishes that if polynomials are dense in weighted spaces for all polynomial degrees, then a smooth weight can be constructed to maintain this density with a controlled lower bound.

Findings

Existence of a smooth weight ensuring polynomial density for all degrees.

Construction of weights that dominate the original weight plus an exponential decay.

Extension of previous density results to smooth weights with specific bounds.

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^{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 $\|f\|_{w} := \sup_{x \in {\mathbb{R}}} w(x)| f( x )|$. In 1998 A.Borichev and M.Sodin completely solved this problem for all those weights $w$ for which ${\mathcal{P} }$ is dense in $ C^{0}_{w} $ but 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 $x\in \mathbb{R}$.