Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator

Abstract

It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue Dm(hβ) of the differential operator d2mdx2m+2dmdxm+1 in the Hilbert space W2(m,0). We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in m=2, we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space W2(2,0) for the case m=2.