A "Weltkonstante" in Time-Frequency Analysis

This project carried out by Markus Faulhuber has two main objectives. The first one is to solve a conjecture formulated by Strohmer and Beaver in 2003 settled in the field of time-frequency analysis. It asks for an optimal sampling strategy, using Gaussian Gabor frames. We explain the concept of Gabor analysis by example. An acoustic signal has a temporal development and a frequency distribution. This means, we can explore how the sinusoids, which define the signal, change over time. From this information, we get a two-dimensional representation of an acoustic signal, similar to a sheet of music. Due to classical uncertainty principles, which appear in the same way in quantum mechanics, we are not able to exactly spot which tone is played at which time. We can rather distinguish that, with a certain probability, certain frequency bands are active during certain time intervals. Using a Gaussian function to localize these joint time-frequency bands, we test at certain points in time how large the amplitudes of certain frequencies are. Since the bands essentially have a circular shape, due to the Gaussian localization, we should test the time-frequency bands in a hexagonal pattern, as this is the most efficient way to spread discs in the plane. The second problem originates from geometric function theory. It was posed in 1929 by Landau and its solution is conjectured to be given in the work by Rademacher in 1943. We are given a disc of radius 1 and we want to map it conformal to a subset of the plane, similar to the way cartographers map the globe on a plane. A theorem by Landau asserts that in the resulting, new object, one can still place a disc of some maximal radius. The problem is to find out how big this radius is at least. Rademacher constructed a tessellation of the disc which he mapped to a hexagonal tessellation of the plane, this means that the plane is covered with equilateral triangles and their vertices are not part of the tessellation. The largest disc which can be placed in this image is the circumcircle of such a triangle. In this project I want to solve the Strohmer and Beaver conjecture and, as well, show that the problem of optimal Gaussian Gabor frames, which might be seen as something rather applied since they are used in many areas of signal processing, is deeply connected to Landau`s problem, which can be considered as very abstract. The findings of the project will substantially widen and deepen our understanding of Gaussian Gabor frames and their use in digital signal processing.

The project called "A 'Weltkonstante' in Time-Frequency Analysis", led by Dr. Markus Faulhuber established new connections between different fields of mathematics. The main conjecture of the project was shown to be true: Is the optimal sampling strategy for certain digital filter systems related to an abstract, still open problem in geometric function theory? A problem under consideration was to find optimal sampling strategies for Gabor systems with a Gaussian window. Gabor systems (or variants) have, e.g., applications in medical signal processing and were used for the detection of gravitational waves. We have a class of building blocks, in this case sine waves damped by a Gauss function, and we want to efficiently approximate given signals, e.g. sound. Due to the damping, we can only use the wave for a certain time interval and then have to shift it, e.g. by 1 second. Also,we need to use waves with different frequencies, say 1Hz, 2Hz, and so on. By combining the time- and the frequency-shifts, we can draw a picture (seconds on the x-axis, Hz on the y-axis), similar to a musical score. Intuitively, the points of the sampling pattern should not interact too much, but should also not leave too big holes. By the theory of disc packing and covering, the hexagonal lattice provides both features. There is no reasonable doubt that a hexagonal sampling pattern is optimal, but the mathematical proof is missing. The project's main result was showing that the sampling problem is linked to the problem of finding Landau's constant, dating back to 1929. The problem is to find a map with certain properties from the hyperbolic disc to the plane. The hyperbolic disc is an abstract, mathematical object, however with applications. In hyperbolic geometry the sum of angles in a triangle is less than 180 degrees. Such a map must disturb areas or angles or both. A more familiar example is spherical geometry, where angles of triangles sum up to more than 180 degrees. We encounter it on the globe and a map of the earth in an atlas cannot conserve both, angles and areas. The problem for the hyperbolic disc, in a simplified version, is the following; if the measure of disturbance at the origin is fixed, how large can the largest disc be, that can be fully placed in any such mapping? Surprisingly, this rather abstract problem and the problem of finding an optimal sampling strategy have a lot in common and their expected solutions are the same. Also, a number of other unsolved problems, e.g., the process of crystallization, are mathematically not different from the above problems. This is future research to be carried out.