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.

我们根据形成框架的函数来考虑希尔伯特空间上有界线性算子的分解。与奇异值分解类似，由此产生的框架分解对问题的结构和不适定性信息进行编码，可用作设计和实现有效数值求解方法的基础。与奇异值分解相反，所提出的框架分解可以明确地推导出为一大类算子，特别是那些满足特定稳定性条件的算子。为了展示这种方法的实用性，我们考虑了来自断层扫描领域的不同例子。