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

TL;DR

This paper introduces explicit frame decompositions for bounded linear operators in Hilbert spaces, providing insights into their structure and ill-posedness, with applications demonstrated in tomography.

Contribution

It presents a novel explicit frame decomposition method for operators, extending the classical singular-value decomposition, applicable to a broad class of operators with stability conditions.

Findings

Frame decompositions encode operator structure and ill-posedness.

Explicit decompositions are derived for operators satisfying stability conditions.

Applications in tomography demonstrate the practical usefulness of the approach.

Abstract

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.