Exponentially weighted Polynomial approximation for absolutely continuous functions

Source abstract

We discuss a polynomial approximation on $\mathbb{R}$ with a weight $w$ in $\mathcal{F}(C^{2} +)$ (see Section 2). The de la Vallée Poussin mean $v_n(f)$ of an absolutely continuous function $f$ is not only a good approximation polynomial of $f$, but also its derivatives give an approximation for the derivative $f'$. More precisely, for $1 \leq p \leq \infty$, we have $\lim_{n \rightarrow \infty}\|(f - v_{n}(f))w\|_{L^{p}(\mathbb{R})} =0$ and $\lim_{n \rightarrow \infty}\|(f' - v_{n}(f)')w\|_{L^{p}(\mathbb{R})} =0$ whenever $f''w \in L^{p}(\mathbb{R})$.