### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 7 (2017)

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

### UDK: 517.912

### Pages: 29 - 39

### DOI: 10.31029/demr.7.3

We consider the system of functions ${\cal \psi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ 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 orthonor-\linebreak mal system of functions ${\cal\psi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series and Fourier sums by the system
${\cal\psi}_{r,n}(x)$ $(r = 1,2, \ldots, n = 0,1, \ldots)$ are convenient and a very effective tool for the approximate solution of the Cauchy problem for difference equations.

**Keywords: **
Sobolev orthogonal functions, functions orthogonal on the grid, approximation of discrete functions, mixed series by the functions orthogonal on a uniform grid, iterative process for the approximate solution of difference equations.