Geodesic
Best quadrature formulas on classes of differentiable periodic functions
Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 475-481

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A solution is given to the problem of finding the best quadrature formula among formulas of the form $$ \int_0^{2\pi}f(x)\,dx\approx\sum_{k=0}^{m-1}\sum_{l=0}^\rho p_{k,l}f^{(l)}(x_k) $$ which are exact in the case of a constant, for $\rho=r-1$, $r=1,2,3,\dots$ and $\rho=r-2$, $r$ even, for the classes $W^{(r)}L_qM$ of $2\pi$-periodic functions. 
@article{MZM_1969_6_4_a12,
     author = {N. E. Lushpai},
     title = {Best quadrature formulas on classes of differentiable periodic functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {475--481},
     publisher = {mathdoc},
     volume = {6},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a12/}
}
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N. E. Lushpai. Best quadrature formulas on classes of differentiable periodic functions. Matematičeskie zametki, Tome 6 (1969) no. 4, pp. 475-481. http://geodesic.mathdoc.fr/item/MZM_1969_6_4_a12/