Abstract:
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.
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