In 1924 S. Bernstein [Bull. Soc. Math. France 52 (1924), 399-410] asked for conditions on a uniformly bounded
R
\mathbb {R}
Borel function (weight)
w
:
R
→
[
0
,
+
∞
)
w: \mathbb {R} \to [0, +\infty )
which imply the denseness of algebraic polynomials
P
{\mathcal {P} }
in the seminormed space
C
w
0
C^{\,0}_{w}
defined as the linear set
{
f
∈
C
(
R
)
|
w
(
x
)
f
(
x
)
→
0
as
|
x
|
→
+
∞
}
\{f \in C (\mathbb {R}) \ | \ w (x) f (x) \to 0 \ \mbox {as} \ {|x| \to +\infty }\}
equipped with the seminorm
‖
f
‖
w
:=
sup
x
∈
R
w
(
x
)
|
f
(
x
)
|
\|f\|_{w} := \sup \nolimits _{x \in {\mathbb {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
w
for which
P
{\mathcal {P} }
is dense in
C
w
0
C^{\,0}_{w}
but for which there exists a positive integer
n
=
n
(
w
)
n=n(w)
such that
P
{\mathcal {P} }
is not dense in
C
(
1
+
x
2
)
n
w
0
C^{\,0}_{(1+x^{2})^{n} w}
. In the present paper we establish that if
P
{\mathcal {P} }
is dense in
C
(
1
+
x
2
)
n
w
0
C^{\,0}_{(1+x^{2})^{n} w}
for all
n
≥
0
n \geq 0
, then for arbitrary
ε
>
0
\varepsilon > 0
there exists a weight
W
ε
∈
C
∞
(
R
)
W_{\varepsilon } \in C^{\infty } (\mathbb {R})
such that
P
{\mathcal {P}}
is dense in
C
(
1
+
x
2
)
n
W
ε
0
C^{\,0}_{(1+x^{2})^{n} W_{\varepsilon }}
for every
n
≥
0
n \geq 0
and
W
ε
(
x
)
≥
w
(
x
)
+
e
−
ε
|
x
|
W_{\varepsilon } (x) \geq w (x) + \mathrm {e}^{- \varepsilon |x|}
for all
x
∈
R
x\in \mathbb {R}
.