EXPONENTIALLY WEIGHTED POLYNOMIAL APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS

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 $f^{\prime}$. 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 $\lim_{n \rightarrow \infty}\|(f^{\prime} - v_{n}(f)^{\prime}) w\|_{L^{p}(\mathbb{R})}=0$ whenever $f^{\prime\prime} w \in L^{p}(\mathbb{R})$.