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2015 Rapid polynomial approximation in $\boldsymbol{L_2}$-spaces with Freud weights on the real line
Rui Xie, Marcel de Jeu
Banach J. Math. Anal. 9(1): 216-234 (2015). DOI: 10.15352/bjma/09-1-16

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.

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Rui Xie. Marcel de Jeu. "Rapid polynomial approximation in $\boldsymbol{L_2}$-spaces with Freud weights on the real line." Banach J. Math. Anal. 9 (1) 216 - 234, 2015. https://doi.org/10.15352/bjma/09-1-16

Information

Published: 2015
First available in Project Euclid: 19 December 2014

zbMATH: 1337.46029
MathSciNet: MR3296096
Digital Object Identifier: 10.15352/bjma/09-1-16

Subjects:
Primary: 46E35
Secondary: 41A10 , 41A25

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

Rights: Copyright © 2015 Tusi Mathematical Research Group

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