  The representations of the kernels of the transmutation operator and of its
inverse relating the one-dimensional Schr\"odinger operator with the second
derivative are obtained in terms of the eigenfunctions of a corresponding
Sturm-Liouville problem. Since both series converge slowly and in general only
in a certain distributional sense we find a way to improve these expansions and
make them convergent uniformly and absolutely by adding and subtracting
corresponding terms. A numerical illustration of the obtained results is given.

Classical Analysis and ODEs;Analysis of PDEs;Numerical Analysis