Systems of functions orthogonal in the sense of Sobolev associated with Haar functions and the Cauchy problem for ODEs

We consider systems of functions ${\mathcal{X}}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$, generated by Haar functions $\chi_{n}(x)$ $(n=1,2,\ldots)$, that form the Sobolev orthonormal system with respect to the scalar product of the following form $<f,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_{0}^{1}f^{(r)}(t)g^{(r)}(x)dx$. It is shown that the Fourier series and sums with respect to the system ${\mathcal{X}}_{r,n}(x)$ $(n=0,1,\ldots)$ are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).