The Institute of Acoustics (ARI) completed the two-year project “Frames and Unbounded Operator (FUn)”: in August 2022. The project was funded by the Innovation Fund „Research, Science and Society“ of the Austrian Academy of Sciences, led by Peter Balazs financing Mitra Shamsabadi and Konrad Schrempf (at the ARI until 2021).
Frame theory as an active field in mathematics and applications (i.e. in signal processing) was established as a strong tool for working with bounded operators. Unbounded operators appear, for example, in physics and are a more delicate but interesting subdomain of operator theory.
The contribution of this Project “FUn” is three-fold:
First, we have combined these approaches and link frame theory to unbounded operators. First, frames have gained a lot of attention for their ability to represent signals and still allow a lot of freedom. Recently, this has also been used for operators – for example for discretizing physical systems. Here, we used this to describe and represent unbounded operators, motivated by the fact that this question arose in quantum physics. We, therefore, tackled and partly solved a problem of John von Neuman, open for nearly a hundred years now.
Secondly, we have gone beyond that and have taken a closer look at how to split a large system of data into several smaller systems. This is naturally fit to the concept of fusion frames, which are subspaces that fulfil a kind of frame-like inequality. Precisely, we have combined the fusion frame theory and unbounded operators, amongst other investigations.
Finally, frame theory also defines and investigates other kinds of sequences. Frames lead to bounded operators, in a canonical way. However, there are other sequences that are – at least from a theoretical point of view – interesting. Among those, within this project we can investigate so called lower frame and Riesz-Fischer sequences and discuss their inter-relation. With our approach, we can show that – in some sense – canonical reconstruction is possible.
The results of the project will be published in articles. One has been published in Axioms (see below), four articles have been submitted and further four articles are in preparation. Also, the research was presented at conferences such as at the International Workshop on Operator Theory and its Applications (IWOTA2022), at Applied Harmonic Analysis and Friends (Strobl 2022) and at the 8th International Conference on Computational Harmonic Analysis (ICCHA 2022).
Publication
Balazs, P.; Bellomonte, G.; Hosseinnezhad, H.: "Frame-related Sequences in Chains and Scales of Hilbert Spaces", Axioms, Vol. 11(4), Special Issue: Time-Frequency Analysis, Distributions, and Operators (link to article)