One property of best quadrature formulas

It is established that for class $W_p^r$ $(r=1,2,\dots;1\le p\le\infty)$ the best quadrature formulas of the form
\begin{gather*} \int_0^1f(x)\,dx=\sum_{k=0}^\rho\sum_{i=1}^na_{ik}f^{(k)}(x_i)+R(f) \\ (0\le\rho\le r-1) \end{gather*}
when $\rho=2m$ and $\rho=2m+1$ coincide with one another. This same fact also supervenes for the class $\widetilde{W}_p^r$ ($r=1,2,\dots$; $1\le p\le\infty$) of periodic functions.