Smoothing of weights in the Bernstein approximation problem

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:= 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 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 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.