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

Abstract

Abstract 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 β ) $D_{m}(h\beta )$ of the differential operator d 2 m d x 2 m + 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.

Keywords

• Differential operator
• Discrete analogue
• Hilbert space
• Discrete argument functions
• Optimal quadrature formula