Sobolev orthogonal functions on the grid, generated by discrete orthogonal functions and the Cauchy problem for the difference equation

We consider the system of functions ${\psi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ orthonormal on Sobolev with respect to the inner product of the form $\langle f,g\rangle=\sum_{k=0}^{r-1}\Delta^kf(0)\Delta^kg(0)+ \sum_{j=0}^\infty\Delta^rf(j)\Delta^rg(j)\rho(j)$, generated by a given orthonormal system of functions ${\psi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series and Fourier sums by the system ${\psi}_{r,n}(x)$ $(r = 1,2, \ldots, n = 0,1, \ldots)$ are convenient and a very effective tool for the approximate solution of the Cauchy problem for difference equations.