The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

Abstract

The discrete analogue of the differential operator plays an important role in the construction of interpolation, quadrature and cubature formulas. In this work, we consider a discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers $m$. The effectiveness of this operator by constructing an optimal quadrature formula in the space $L_2^{(2,0)}(0,1)$ is shown. The errors of the optimal quadrature formula in $W_2^{(2,1)}(0,1)$ space and the optimal quadrature formula in $L_2^{(2,0)}(0,1)$ space are numerically compared. Numerical results showed that the error of the optimal quadrature formula what constructed in this work is smaller than the error of the quadrature formula constructed in $W_2^{(2,1)}(0,1)$ space. MSC: 41A05, 41A15, 65D30, 65D32, 65M60, 65L60.