Localization (of) Operators and Operator Reconstruction

In quantum mechanics, Heisenberg`s uncertainty principle states that one cannot simultaneously measure the position and momentum of a particle. Mathematically, this uncertainty relation can be expressed by stating that a function and its Fourier transform cannot both have compact support or decay arbitrarily quickly to zero. Over the decades, various versions of this principle have been proven. Particularly important are the so-called localization operators, which, although investigated since the 1960s, remain an active area of research today. A localization operator restricts a function to certain regions in time and frequency space. These operators have found diverse applications, from sampling theory to statistics. One aim of this project is to investigate such uncertainty relations for operators instead of functions. In mathematics, an operator is a linear mapping between two spaces of functions or vectors. These mappings are used, for example, in mobile communications to model a communication channel. It is not immediately clear why operators are subject to uncertainty principles. However, an examination using methods of quantum harmonic analysis shows that this is indeed the case. This field attempts to establish parallel results of harmonic analysis for functions also for operators. We will use methods of quantum harmonic analysis and the theory of localization operators to conduct a systematic study of operator localization. The second aim of this project is to examine how characteristics of an operator can be reconstructed solely based on one (or a few) measurements of an operator`s output. Operator reconstruction could, for example, involve identifying the current calibration of a hearing aid or detecting an object using radar. Classical reconstruction methods aim to restore signals or operators as accurately as possible. However, this is often unnecessary, as certain parameters often already contain the essential information about the behavior of an operator and it will be part of this project to investigate when this leads to satisfactory reconstruction. A particular focus will be on whether the localization properties of an operator facilitate its reconstruction. We will draw on the theory of operator identification and use randomized methods where Gaussian noise serves as an input function for an operator. Such statistical algorithms are playing an increasingly important role in applied mathematics, as they provide reliable results despite their inherent randomness, while following a simple structure.