Smoothing of weights in the Bernstein approximation problem

Andrew Bakan, Jürgen Prestin


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 C0 w defined as the linear set {f ∈ C(R) | w(x)f(x) → 0 as |x| →+∞} equipped with the seminorm ‖f‖w:= supx ∈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 C0 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 Wε ∈ C (R) such that P is dense in (Formula Presented) for every n ≥ 0 and Wε(x) ≥ w(x) +e−ε|x| for all x ∈ R.

Original languageEnglish
JournalProceedings of the American Mathematical Society
Issue number2
Pages (from-to)653-667
Number of pages15
Publication statusPublished - 2017


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