We study numerical methods for the solution of general linear moment problems,
where the solution belongs to a family of nested subspaces of a Hilbert space. Multi-level
algorithms, based on the conjugate gradient method and the Landweber--Richardson method are
proposed that determine the "optimal" reconstruction level a posteriori from quantities
that arise during the numerical calculations. As an important example we discuss the
reconstruction of band-limited signals from irregularly spaced noisy samples, when the
actual bandwidth of the signal is not available. Numerical examples show the usefulness of
the proposed algorithms.