## Banach Journal of Mathematical Analysis

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

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

Rui Xie and Marcel de Jeu

#### 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

**Permanent link to this document**

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

**Subjects**

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

**Keywords**

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

#### 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