For a compact set
E
E
with connected complement, let
A
(
E
)
A(E)
be the uniform algebra of functions continuous on
E
E
and analytic interior to
E
.
E.
We describe
A
(
E
,
W
)
,
A(E,W),
the set of uniform limits on
E
E
of sequences of the weighted polynomials
{
W
n
(
z
)
P
n
(
z
)
}
n
=
0
∞
,
\{W^n(z)P_n(z)\}_{n=0}^{\infty },
as
n
→
∞
,
n \to \infty ,
where
W
∈
A
(
E
)
W \in A(E)
is a nonvanishing weight on
E
.
E.
If
E
E
has empty interior, then
A
(
E
,
W
)
A(E,W)
is completely characterized by a zero set
Z
W
⊂
E
.
Z_W \subset E.
However, if
E
E
is a closure of Jordan domain, the description of
A
(
E
,
W
)
A(E,W)
also involves an inner function. In both cases, we exhibit the role of the support of a certain extremal measure, which is the solution of a weighted logarithmic energy problem, played in the descriptions of
A
(
E
,
W
)
.
A(E,W).