  The spectral data of a vibrating string are encoded in its so-called
characteristic function. We consider the problem of recovering the distribution
of mass along the string from its characteristic function. It is well-known
that Stieltjes' continued fraction provides a solution of this inverse problem
in the particular case where the distribution of mass is purely discrete. We
show how to adapt Stieltjes' method to solve the inverse problem for a related
class of strings. An application to the excursion theory of diffusion processes
is presented.
