### Approximation theory

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

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

### UDK: 517.912

### Pages: 77 - 85

### DOI: 10.31029/demr.7.9

We consider the system of functions ${\cal \psi}_{1,n}(x, N)$ ($n=0,1,\ldots,$ $N$), orthonormal in Sobolev sense and generated by a given orthonormal on finite grid $\Omega_N=\left\{ 0,1,\ldots,N-1 \right\}$ system of functions ${\cal\psi}_{n}(x,N)$ $( n=0,1,\ldots,N-1)$.
These new functions are orthonormal with respect to the inner product of the following type: $\langle f,g\rangle = f(0)g(0)+
\sum_{j=0}^{N-1}\Delta f(j)\Delta g(j)\rho(j)$.
It is shown that the finite Fourier series by the functions
${\cal\psi}_{1,n}(x)$ and their partial sums are convenient and a very effective tool for the approximate solution of the Cauchy problem for nonlinear difference equations.

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