Extension of the Euler-Maclaurin quadrature formula in a Hilbert space

In the paper we consider an extension problem of the Euler-Maclaurin quadrature formula in the Hilbert space  by constructing an optimal quadrature formula. The optimal quadrature formula is obtained by minimizing the error of the formula by coefficients at values of the second derivative of a integrand. Using the discrete analogue of the operator. The explicit formulas for the coefficients of the optimal quadrature formula are obtained. Furthermore, it is proved that the obtained quadrature formula is exact for any function of the set.. Finally, the square of the norm of the error functional for the constructed quadrature formula is calculated. It is shown that the error of the obtained optimal quadrature formula is less than the error of the classical Euler-Maclaurin quadrature formula on the space.

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