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} .