## Abstract

In 1924 S. Bernstein [Bull. Soc. Math. France 52 (1924), 399-410] asked for conditions on a uniformly bounded R Borel function (weight) w: R → [0, +∞) which imply the denseness of algebraic polynomials P in the seminormed space C^{0} _{w} defined as the linear set {f ∈ C(R) | w(x)f(x) → 0 as |x| →+∞} equipped with the seminorm ‖f‖_{w}:= sup_{x} _{∈R} w(x)|f(x)|. 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 P is dense in C^{0} _{w} but for which there exists a positive integer n = n(w) such that P is not dense in (Formula presented). In the present paper we establish that if P is dense in (Formula presented) for all n ≥ 0, then for arbitrary ε > 0 there exists a weight W_{ε} ∈ C^{∞} (R) such that P is dense in (Formula Presented) for every n ≥ 0 and W_{ε}(x) ≥ w(x) +e^{−ε|x|} for all x ∈ R.

Original language | English |
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Journal | Proceedings of the American Mathematical Society |

Volume | 146 |

Issue number | 2 |

Pages (from-to) | 653-667 |

Number of pages | 15 |

ISSN | 0002-9939 |

DOIs | |

Publication status | Published - 2017 |