Banach Journal of Mathematical Analysis

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

Rui Xie and Marcel de Jeu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 19 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 41A10: Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10} 41A25: Rate of convergence, degree of approximation

Weighted $L_2$-space Freud weight rapid polynomial approximation Jackson inequality Markov inequality


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.

Export citation