Geodesic
Best quadrature formula on the class $W_*^rL_2$
Matematičeskie zametki, Tome 16 (1974) no. 2, pp. 193-204 
N. E. Lushpai. Best quadrature formula on the class $W_*^rL_2$. Matematičeskie zametki, Tome 16 (1974) no. 2, pp. 193-204. http://geodesic.mathdoc.fr/item/MZM_1974_16_2_a1/
@article{MZM_1974_16_2_a1,
     author = {N. E. Lushpai},
     title = {Best quadrature formula on the class $W_*^rL_2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {193--204},
     year = {1974},
     volume = {16},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_2_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

For the classes of periodic functions with $r$-th derivative integrable in the mean,we obtain a best quadrature formula of the form \begin{gather*} \int_0^1f(x)\,dx=\sum_{k=0}^{m-1}\sum_{l=0}^{\rho}p_{k,l}f^{(l)}(x_k)+R(f),\quad0\le\rho\le r-1, \\ 0\le x_0<x_1<\dots<x_m-1\le1, \end{gather*} where $\rho=r-2$ and $r-3$, $r=3,5,7,\dots$, and we determine an exact bound for the error of this formula.