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This paper is devoted to the process of finding the upper bound for the absolute error of the optimal quadrature formula in the space W2​​(m,m−1) of real-valued, periodic functions. For this the extremal function of the quadrature formula is used. In addition, it is shown that the norm of the error functional for the optimal quadrature formula constructed in the space W2​​(m,m−1)is less than the value of the norm of the error functional for the optimal quadrature formula in the Sobolev space L2​​(m).

In the present paper in L 2 ( m ) ( 0 , 1 ) space the optimal quadrature formulas with derivatives are constructed for approximate calculation of the Cauchy type singular integral. Explicit formulas for the optimal coefficients are obtained. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L 2 ( m ) ( 0 , 1 ) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number N and for any [ m 2 ] ≥ s ≥ 1 using Sobolev method which is based on discrete analogue of the differential operator d 2 m /d x 2 m . In particular, for s = [ m / 2 ] optimality of the classical Euler–Maclaurin quadrature formula is obtained. Starting from [ m /2] > s new optimal quadrature formulas are obtained. Finally, numerical results which confirm theoretical are presented. [ABSTRACT FROM AUTHOR]

Abstract: This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space . In this paper the quadrature sum consists of values of the integrand at nodes and values of the first derivative of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number and for any using the S.L. Sobolev method which is based on a discrete analog of the differential operator . In particular, for optimality of the classical Euler–Maclaurin quadrature formula is obtained. Starting from new optimal quadrature formulas are obtained. [Copyright &y& Elsevier]

Abstract: In this paper there is considered the problem of construction of a new optimal quadrature formula in the sense of Sard in Hilbert space, using S.L. Sobolevʼs method. There are given explicit formulas for coefficients of the optimal quadrature formula. Furthermore some numerical results are presented. The constructed optimal quadrature formula is exact for functions and . [Copyright &y& Elsevier]