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.

Localized, Fusion and Tensors of Frames Modern technologies-from medical imaging and wireless communication to audio processing and machine learning-depend on turning complex mathematical relations into forms that computers can use efficiently and reliably. Our project develops better ways to represent these rules, known as "operators," so that computations become faster, more stable, and more accurate, even when data are noisy or incomplete. Traditionally, operators are represented using orthonormal bases, like the coordinate grids learned in school. While elegant, these can be too rigid for real-world data. Frames offer a more flexible alternative: they allow redundancy, which is not wasteful but protective. Redundancy stabilizes calculations and supports robust recovery of information when parts of the data are missing or corrupted. We focus on three powerful families of frames: * Localized frames: Elements correlate mostly with "nearby" elements. This can lead to sparse, well-structured matrices and faster algorithms. * Fusion frames: Information is organized by subspaces or subsets. This can be ideal for distributed sensing, parallel computing, and combining heterogeneous data. * Tensor products of frames: Higher-dimensional tools are built from lower-dimensional pieces, essential for images, video, and multi-sensor systems. Key results and contributions: - Unified notion of localization: We extended the concept of "local interactions" to generalized frames, fusion frames, and tensor products. This provides a common language to be able to design accurate and efficient matrix representations across many problem types. - Foundations for fusion-frame operator representations: We developed the core theory to represent operators using fusion frames, enabling stable "divide-and-conquer" methods that could split large problems into smaller parts computable in parallel. - Bridging frame theory, signal processing and machine learning: We connected frame theory with filter bank techniques (central in audio/image compression and analysis) and explored links to advance machine learning. These insights support more interpretable and robust models for high-dimensional data. - Closing a theoretical gap: We advanced the theory of dual continuous frames with analyzing tensor products in this setting, strengthening tools used in physics. - Community resource: We prepared a comprehensive survey on fusion frames to help researchers and practitioners adopt these methods. Better operator representations translate into tangible benefits-faster computations, reduced memory needs, resilience to noise, and improved accuracy. The outcomes of this project can enhance future applications such as medical image reconstruction, speech enhancement, wireless networks, and trustworthy machine learning. In short, Localized, Fusion and Tensors of Frames delivers mathematical tools that can turn complex computations into reliable, practical solutions-advancing the possibilities to transform raw data into clear, actionable information across science and engineering.