## Banach Journal of Mathematical Analysis

### Rapid polynomial approximation in $\boldsymbol{L_2}$-spaces with Freud weights on the real line

#### Abstract

The weights $W_{\alpha}(x)=\mathrm{exp}(-|x|^{\alpha})$ $(\alpha>1)$ form a subclass of Freud weights on the real line. Primarily from a functional analytic angle, we investigate the subspace of $L_{2}(\mathbb{R}, W_{\alpha}^{2} dx)$ consisting of those elements that can be rapidly approximated by polynomials. This subspace has a natural Fr\'echet topology, in which it is isomorphic to the space of rapidly decreasing sequences. We show that it consists of smooth functions and obtain concrete results on its topology. For $\alpha=2$, there is a complete and elementary description of this topological vector space in terms of the Schwartz functions.

#### Article information

Source
Banach J. Math. Anal., Volume 9, Number 1 (2015), 216-234.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.bjma/1419000588

Digital Object Identifier
doi:10.15352/bjma/09-1-16

Mathematical Reviews number (MathSciNet)
MR3296096

Zentralblatt MATH identifier
1337.46029

#### Citation

Xie, Rui; de Jeu, Marcel. Rapid polynomial approximation in $\boldsymbol{L_2}$-spaces with Freud weights on the real line. Banach J. Math. Anal. 9 (2015), no. 1, 216--234. doi:10.15352/bjma/09-1-16. https://projecteuclid.org/euclid.bjma/1419000588