Fix a positive integer
n
n
and
1
>
p
>
∞
1>p>\infty
. We provide expressions for the weighted
L
p
L^{p}
distance
\[
inf
f
∫
0
2
π
|
1
−
f
|
p
w
d
λ
,
\inf _{f} \int ^{2 \pi }_{0} | 1 - f |^p w\,d\lambda ,
\]
where
d
λ
d\lambda
is normalized Lebesgue measure on the unit circle,
w
w
is a nonnegative integrable function, and
f
f
ranges over the trigonometric polynomials with frequencies in
\[
S
1
=
{
…
,
−
3
,
−
2
,
−
1
}
∪
{
1
,
2
,
3
,
…
,
n
}
,
S_1 = \{ \ldots , -3, -2, -1\}\cup \{ 1, 2, 3,\ldots , n\},
\]
\[
S
2
=
{
…
,
−
3
,
−
2
,
−
1
}
∖
{
−
n
}
,
S_2=\{ \ldots , -3, -2,-1\}\setminus \{-n\},
\]
or
\[
S
3
=
{
…
,
−
3
,
−
2
,
−
1
}
∪
{
n
}
.
S_3 =\{ \ldots , -3, -2, - 1\}\cup \{n\}.
\]
These distances are related to other extremal problems, and are shown to be positive if and only if
log
w
\log w
is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with
w
w
.