### The best, by coefficients, weighted quadrature formulas of the form $$$\sum\limits_{k=1}^n \sum\limits_{i=0}^{r-1} A_{ki} f^{(i)}_{jk}$$$ for some classes of differentiable functions

#### Abstract

For the quadrature formula (with non-negative, integrable on $$$[0,1]$$$ function) that is defined by the values of the function and its derivatives of up to and including $$$(r-1)$$$-th order, we find the form of the best coefficients $$$A^0_{ki}$$$ ($$$k = \overline{1, n}$$$, $$$i = \overline{0, r-1}$$$) for fixed nodes $$$\gamma_k$$$ ($$$k = \overline{1, n}$$$) and we give the sharp estimate of the remainder of this formula on the classes $$$W^r_p$$$, $$$r = 1, 2, \ldots$$$, $$$1 \leqslant p \leqslant \infty$$$.

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Nikolskii S.M. *Quadrature formulas*, 1974. (in Russian)

Lebed' G.K. "On quadrature formulas with the smallest estimate of remainder on some classes of functions", *Izv. AN SSSR. Ser. Matem.*, 1970; 3(4): pp. 645-669. (in Russian) doi:10.1070/IM1970v004n03ABEH000925

Lushpai N.Ye., Busarova T.N. "Optimal formulas of numerical integration for some classes of differentiable functions", *Ukrainian Math. J.*, 1978; 30(2): pp. 234-238. (in Russian) doi:10.1007/BF01085641

DOI: https://doi.org/10.15421/248707

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