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

We consider systems of functions phi(r,n) (x) (r = 1, 2, ..., n = 0, 1, ... ) that are Sobolev-orthonormal with respect to a scalar product of the form < f, g > = Sigma(r - 1)(nu = 0) f((nu)) (a)g((nu))(a) + integral(b)(a) f((r)) (x)g((r)) (x)rho(x) dx and are generated by a given orthonormal system of functions phi(n) (x) (n = 0, 1, ... ). The Fourier series and sums with respect to the system phi(r,n) (x) (r = 1, 2, ..., n = 0,1, ... ) are shown to be a convenient and efficient tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).