  It is shown that eigenvalues of Laplace-Beltrami operators on compact
Riemannian manifolds can be determined as limits of eigenvalues of certain
finite-dimensional operators in spaces of polyharmonic functions with
singularities. In particular, a bounded set of eigenvalues can be determined
using a space of such polyharmonic functions with a fixed set of singularities.
It also shown that corresponding eigenfunctions can be reconstructed as uniform
limits of the same polyharmonic functions with appropriate fixed set of
singularities.
