  We consider the decomposition of bounded linear operators on Hilbert spaces
in terms of functions forming frames. Similar to the singular-value
decomposition, the resulting frame decompositions encode information on the
structure and ill-posedness of the problem and can be used as the basis for the
design and implementation of efficient numerical solution methods. In contrast
to the singular-value decomposition, the presented frame decompositions can be
derived explicitly for a wide class of operators, in particular for those
satisfying a certain stability condition. In order to show the usefulness of
this approach, we consider different examples from the field of tomography.
