We prove the uniqueness of the quadrature formula with minimal error in the space
W
~
q
r
[
a
,
b
]
,
1
>
q
>
∞
\tilde W_q^r[a,b],1 > q > \infty
, of
(
b
−
a
)
(b - a)
-periodic differentiable functions among all quadratures with n free nodes
{
x
k
}
1
n
\{ {x_k}\} _1^n
,
a
=
x
1
>
⋯
>
x
n
>
b
a = {x_1} > \cdots > {x_n} > b
, of fixed multiplicities
{
v
k
}
1
n
\{ {v_k}\} _1^n
, respectively. As a corollary, we get that the equidistant nodes are optimal in
W
~
q
r
[
a
,
b
]
\tilde W_q^r[a,b]
for
1
⩽
q
⩽
∞
1 \leqslant q \leqslant \infty
if
v
1
=
⋯
=
v
n
{v_1} = \cdots = {v_n}
.