One new method for constructing compound optimal quadrature formulas

In the monographs of S.L. Sobolev and Kh.M. Shadimetov proved that the rectangle formula is an optimal quadrature formula in the Sobolev space of periodic functions. In addition, it is shown there that the error of the constructed optimal quadrature formula is estimated in terms of the step of the quadrature formula and Bernoulli numbers. The subject of this work is the construction of an optimal composite quadrature formula, i.e. quadrature formula involving the values of the derivatives of the function being integrated. Here we propose a new method for constructing composite optimal quadrature formulas based on the periodization of an arbitrary function from the Sobolev space of non-periodic functions and optimal quadrature formulas for rectangles.

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