01.06.2022

# Continuous Frames and Tensors

Peter Balazs, director at the Acoustics Research Institute, gives an insight in his recently published research about continuous frames in tensor product Hilbert spaces. In applications, it is often beneficial to represent data not in raw form, but in a representation that more readily reveals important structure in and properties of said data. In many cases, such representations further benefit from redundancy, that is, they expand said data into a larger number of coefficients, compared to the original data points. As an example, let us mention a time-frequency representation via a short-time Fourier transform that works with a 75% overlap of the windows.

This results in a redundancy of 4, leading to advantages, for example, in error-correction. The mathematical properties of redundant representations, so-called frames, are studied within frame theory, which has had significant influence in various areas of mathematics, signal processing, and physics. Frame theory answers questions related to decompositions of a signal into atoms, or to numerical properties of related signal representations. While classical frame theory considers discrete collections of atoms, is is entirely possible to forego this restriction and study frames in an analog setting, leading to the so-called continuous frames, where the elements are indexed by a continuous variable. Doing so has enriched even the study of classical representations like the Short-Time Frequency-Transform (STFT), and enabled many novel theoretical results.

Like frames, tensors and tensor products appear in many scientific fields, such as mathematics, physics, and machine learning. They provide a formal tool for the description of linear operations in  higher-dimensional spaces and the construction of operators on said spaces from lower dimensional operations, preserving linearity in each component. Kernel theorems related to tensor products are used to reveal important properties of various operators.

In our work the concepts of continuous frames and tensors are combined for the first time. Among many mathematical results, we show for the first time a full characterization of all dual continuous frames, i.e., those that allow perfect reconstruction. This question was open for three decades. As a consequence of this result we demonstrate that any continuous tensor frame has  dual systems that are not simple tensors.

Furthermore, we consider properties of continuous frame multipliers in the context of tensor products. Familiar localization operators for short-time Fourier and wavelet transforms are examples of such multipliers. We propose an interpretation of certain bilinear localization operators as density operators for composite (bipartite) quantum systems which, in principle, could be used to describe the state of subsystem in a prescribed region of the phase space. See about density matrix

We have recently published these results in the renowned Journal of Physics A: Mathematical and Theoretical.

Balazs, P., Teofanov, N.: "Continuous frames in tensor product Hilbert spaces, localization operators and density operators" in: Journal of Physics A: Mathematical and Theoretical, Vol 55, no 14, 2022 https://iopscience.iop.org/article/10.1088/1751-8121/ac55eb