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

The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog $D_m(h\beta )$ $D_{m}(h\beta )$ of the differential operator $\frac{d^{2m}}{d{x}^{2m}}+1$ $\frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the $L_2^{(2,0)}(0,1)$ $L_{2}^{(2,0)}(0,1)$ space is demonstrated. The errors of the optimal quadrature formula in the $W_2^{(2,1)}(0,1)$ $W_{2}^{(2,1)}(0,1)$ space and in the $L_2^{(2,0)}(0,1)$ $L_{2}^{(2,0)}(0,1)$ space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the $W_2^{(2,1)}(0,1)$ $W_{2}^{(2,1)}(0,1)$ space.