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 $$$W^r_p$$$, $$$r = 1, 2, \ldots$$$, $$$1 \leqslant p \leqslant \infty$$$.