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We consider the decomposition of bounded linear operators on Hilbert spacesin terms of functions forming frames. Similar to the singular-valuedecomposition, the resulting frame decompositions encode information on thestructure and ill-posedness of the problem and can be used as the basis for thedesign and implementation of efficient numerical solution methods. In contrastto the singular-value decomposition, the presented frame decompositions can bederived explicitly for a wide class of operators, in particular for thosesatisfying a certain stability condition. In order to show the usefulness ofthis approach, we consider different examples from the field of tomography.

Frame Decompositions of Bounded Linear Operators in Hilbert Spaces with Applications in Tomography