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

Abstract

In the present paper we construct the discrete analogue $D_m(hβ)$ of the differential operator $\frac{d^{2m}}{d x^{2m}} + 2ω^2\frac{d^{2m-2}}{d x^{2m-2}} + ω^4\frac{d^{2m-4}}{d x^{2m-4}}$. The discrete analogue $D_m(hβ)$ 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.