Packing, Covering and Time-Frequency Analysis

The project Packing, Covering, and Time-Frequency Analysis, led by Dr. Markus Faulhuber, investigates new mathematical approaches for the digital transmission of signals. The project is a basic research project in mathematics, motivated by practical questions. A main task is to study connections between sampling theorems and discrete geometry. The core problem is as follows: we are given some values of a function and want to know whether these already determine the function completely. The mathematical theory, called sampling theory, is already about one century old. It gained practical relevance when the computers became more powerful and is a central part of our digital communication or medical imaging. While many of the foundational questions have been answered in the last couple of decades, still many mathematical problems remain unsolved. Our focus is on certain structured function systems (Gabor systems) which have been introduced in the 1940s by Dennis Gabor for the purpose of digital communication and who later won the Nobel price in physics. Within the project we want to pursue the question whether symmetries and patterns influence the stability of the transmission and the quality of the reconstruction of signals. In particular, we will study certain examples and want to show in a mathematical rigorous manner that this is the case, with special focus on packing and covering properties of the pattern. For the first time the quality of periodic patterns, which are not lattices, will be studied. A lattice has the special property that if two vectors of the lattice are added, then we obtain another point from the lattice. Many periodic and symmetric patterns do not posses this nice mathematical property, complicating the exact mathematical determination of the quality constants. We will take on this endeavor and as a next step we will also carry out a mathematical comparison with the quality of lattices. Another question in the project concerns the number of points at which we need to know the function in order to be able to reconstruct it with a Gabor system. Our focus will be put on the special class of Gabor systems generated by a Hermite function, which posses nice mathematical properties. We want to show that in general less sampling points are needed than currently proven to be sufficient. This reduces the transmission rate while keeping the guarantee of unique reconstruction of a signal.