Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs

We consider systems of functions ${\varphi}_{r,n}(x)$ ($r=1,2,\dots$, $n=0,1,\dots$) that are Sobolev-orthonormal with respect to a scalar product of the form $\langle f,g\rangle= \sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+ \int_{a}^{b}f^{(r)}(x)g^{(r)}(x)\rho(x)\,dx$ and are generated by a given orthonormal system of functions $\varphi_{n}(x)$ ($n=0,1,\dots$). The Fourier series and sums with respect to the system $\varphi_{r,n}(x)$ ($r=1,2,\dots$, $n=0,1,\dots$) are shown to be a convenient and efficient tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).