Localized, Fusion and Tensors of Frames

Representing operators is an important part of mathematics. This is essential for the possibility to treat operator systems numerically, i.e. with a computer. This is done by transforming them to systems of equations, which can be represented by matrices. Traditionally, orthonormal bases (comparable to the coordinate system learned in school) have been used to obtain such representations. Recently frames have entered the picture, that allow redundant representations. The quality of the matrix representation depends crucially on the chosen class of frames. One particularly promising approach, to describe `good` frames, is the concept of localized frames. Those are frames, where the inter-relation is nice. The investigation of representations of operators with those frames has recently been started, supporting the important role of tensor products of frames in those representations. Finally, frame systems of subspaces, also known as fusion frames are naturally linked to domain decomposition methods for the integral representation of operators. Operator representations with those three classes are still a largely unexplored field in frame theory. We will advance the mathematical theory of localized frames, fusion frames, and their tensor products, for the representation of operators, connecting harmonic analysis, and operator theory.