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On the best interval quadrature formulae for classes of differentiable periodic functions. (English) Zbl 1147.41008
The authors discuss the Kolmogorov problems on optimal quadrature formulae for class \(W^{r}F\) of differentiable periodic functions with rearrangement invariant set \(F\) of their derivative of order \(r\). They prove that for any fixed \(h \in [0,\frac{\pi}{n}]\) the interval quadrature formula having equidistant nodes and equal coefficients, \(b_{j}=\frac{2\pi}{n}\) is optimal for the class \(W^{r}F\). To this end a sharp inequality for antiderivatives of rearrangement of averaged monosplines is proved.

41A55 Approximate quadratures
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