A multi-level algorithm for the solution of moment problems

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 bandlimited 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.