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

We consider systems of functions (, ) that are Sobolev-orthonormal with respect to a scalar product of the form and are generated by a given orthonormal system of functions (). The Fourier series and sums with respect to the system (, ) are shown to be a convenient and efficient tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).