  In arXiv:1306.2914 a method for approximate solution of Sturm-Liouville
equations and related spectral problems was presented based on the construction
of the Delsarte transmutation operators. The problem of numerical approximation
of solutions was reduced to approximation of a primitive of the potential by a
finite linear combination of certain specially constructed functions obtained
from the generalized wave polynomials introduced in arXiv:1208.5984 and
arXiv:1208.6166. The method allows one to compute both lower and higher
eigendata with an extreme accuracy. Since the solution of the approximation
problem is the main step in the application of the method, the properties of
the system of functions involved are of primary interest. In arXiv:1306.2914
two basic properties were established: the completeness in appropriate
functional spaces and the linear independence. In this paper we present a
considerably more complete study of the systems of functions. We establish
their relation with another linear differential second-order equation, find out
certain operations (in a sense, generalized derivatives and antiderivatives)
which allow us to generate the next such function from a previous one. We
obtain the uniqueness of the coefficients of expansions in terms of such
functions and a corresponding generalized Taylor theorem. We construct the
invertible integral operators transforming powers of the independent variable
into the functions under consideration and establish their commutation
relations with differential operators. We present some error bounds for the
solution of the approximation problem depending on the smoothness of the
potential and show that these error bounds are close to optimal in order. Also,
we provide a rigorous justification of the alternative formulation of the
proposed method allowing one to make use of the known initial values of the
solutions at an endpoint.
