Let
A
+
=
{
a
=
(
a
n
)
∈
⋂
p
>
1
ℓ
p
:
a
n
>
0
,
∀
n
∈
N
}
\mathcal {A_{+}}=\{a=(a_{n})\in \bigcap _{p>1}\ell _{p}:a_{n}>0,\,\forall n\in \mathbb {N}\}
and let
{
ϕ
j
}
j
=
1
∞
\{\phi _{j}\}_{j=1}^{\infty }
be an enumeration of all normal distributions with mean a rational number and variance
1
n
2
,
n
=
1
,
2
…
\frac {1}{n^{2}},\,n=1,2\dots
. We prove that there exists an
a
∈
A
+
a\in \mathcal {A_{+}}
such that every probability density function, continuous, with compact support in
R
\mathbb {R}
, can be approximated in
L
1
L^{1}
and
L
∞
L^{\infty }
norm simultaneously by the averages
1
∑
j
=
1
n
a
j
∑
j
=
1
n
a
j
ϕ
j
\frac {1}{\sum _{j=1}^{n}a_{j}}\,\sum _{j=1}^{n}a_{j}\phi _{j}
. The set of such sequences is a dense
G
δ
G_{\delta }
set in
A
+
\mathcal {A_{+}}
and contains a dense positive cone.