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 Dₘ(hβ ) D m ( h β ) of the differential operator d²ᵐdx²ᵐ+1 d 2 m d x 2 m + 1 designed specifically for even natural numbers m . The operator’s effectiveness in constructing an optimal quadrature formula in the L₂(2,0)(0,1) L 2 ( 2 , 0 ) ( 0 , 1 ) space is demonstrated. The errors of the optimal quadrature formula in the W₂(2,1)(0,1) W 2 ( 2 , 1 ) ( 0 , 1 ) space and in the L₂(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 W₂(2,1)(0,1) W 2 ( 2 , 1 ) ( 0 , 1 ) space.