Smoothing of weights in the Bernstein approximation problem

In 1924 S.Bernstein asked for conditions on a uniformly bounded on R Borel function (weight) w:R→[0,+∞) which imply the denseness of algebraic polynomials P in the seminormed space C0w defined as the linear set {f∈C(R) | w(x)f(x)→0 as |x|→+∞} equipped with the seminorm ∥f∥w:=supx∈Rw(x)|f(x)|. In 1998 A.Borichev and M.Sodin completely solved this problem for all those weights w for which P is dense in C0w but there exists a positive integer n=n(w) such that P is not dense in C0(1+x2)nw. In the present paper we establish that if P is dense in C0(1+x2)nw for all n≥0 then for arbitrary ε>0 there exists a weight Wε∈C∞(R) such that P is dense in C0(1+x2)nWε for every n≥0 and Wε(x)≥w(x)+e−ε|x| for all x∈R.