The discrete analogue of the differential operator $\frac{d^{2m}}{d{x}^{2m}}+2{\omega }^2\frac{d^{2m-2}}{d{x}^{2m-2}}+{\omega }^4\frac{d^{2m-4}}{d{x}^{2m-4}}$

Abstract

In the present paper we construct the discrete analogue $D_m(h\beta )$ of the differential operator $\frac{d^{2m}}{d{x}^{2m}}+2{\omega }^2\frac{d^{2m-2}}{d{x}^{2m-2}}+{\omega }^4\frac{d^{2m-4}}{d{x}^{2m-4}}$. The discrete analogue $D_m(h\beta )$ plays the main role in construction of optimal quadrature formulas and interpolation splines minimizing the semi-norm in the $K_2(P_m)$ Hilbert space.