Tohoku Mathematical Journal
2018

March
SECOND SERIES VOL. 70, NO. 1

Tohoku Math. J.
70 (2018), 1-15

Title EXPONENTIALLYWEIGHTED POLYNOMIAL APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS

Author Kentaro Itoh, Ryozi Sakai and Noriaki Suzuki

(Received October 27, 2015, revised December 9, 2015)
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 $f''w \in L^{p}(\mathbb{R})$.

Mathematics Subject Classification. Primary 41A17; Secondary 41A10.

Key words and phrases. Weighted polynomial approximation, absolutely continuous function, Erdös type weight, de la Vallée Poussin mean.

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