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Smoothing of weights in the Bernstein approximation problem

Authors: Andrew Bakan and Jürgen Prestin
Journal: Proc. Amer. Math. Soc. 146 (2018), 653-667
MSC (2010): Primary 41A10, 46E30; Secondary 32A15, 32A60
Published electronically: August 30, 2017
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Abstract: In 1924 S. Bernstein [Bull. Soc. Math. France 52 (1924), 399-410] asked for conditions on a uniformly bounded $ \mathbb{R}$ Borel function (weight) $ w: \mathbb{R} \to [0, +\infty )$ which imply the denseness of algebraic polynomials $ {\mathcal {P} }$ in the seminormed space $ C^{\,0}_{w} $ defined as the linear set $ \{f \in C (\mathbb{R}) \ \vert \ w (x) f (x) \to 0 \ $$ \mbox {as} \ {\vert x\vert \to +\infty }\}$ equipped with the seminorm $ \Vert f\Vert _{w} := \sup \nolimits _{x \in {\mathbb{R}}} w(x)\vert f( x )\vert$. In 1998 A. Borichev and M. Sodin [J. Anal. Math 76 (1998), 219-264] completely solved this problem for all those weights $ w$ for which $ {\mathcal {P} }$ is dense in $ C^{\,0}_{w} $ but for which there exists a positive integer $ n=n(w)$ such that $ {\mathcal {P} }$ is not dense in $ C^{\,0}_{(1+x^{2})^{n} w}$. In the present paper we establish that if $ {\mathcal {P} }$ is dense in $ C^{\,0}_{(1+x^{2})^{n} w}$ for all $ n \geq 0$, then for arbitrary $ \varepsilon > 0$ there exists a weight $ W_{\varepsilon } \in C^{\infty } (\mathbb{R})$ such that $ {\mathcal {P}}$ is dense in $ C^{\,0}_{(1+x^{2})^{n} W_{\varepsilon }}$ for every $ n \geq 0$ and $ W_{\varepsilon } (x) \geq w (x) + \mathrm {e}^{- \varepsilon \vert x\vert}$ for all $ x\in \mathbb{R}$.

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Additional Information

Andrew Bakan
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 01601, Ukraine

Jürgen Prestin
Affiliation: Institut für Mathematik, Universität zu Lübeck, D-23562 Lübeck, Germany

Keywords: Polynomial approximation, weighted approximation, $C^0_{w}$-spaces, entire functions
Received by editor(s): November 20, 2016
Received by editor(s) in revised form: March 22, 2017
Published electronically: August 30, 2017
Additional Notes: This work was completed during the first author’s visit to the University of Lübeck, supported by the German Academic Exchange Service (DAAD, Grant 57210233).
Communicated by: Walter Van Assche
Article copyright: © Copyright 2017 American Mathematical Society

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