Time-frequency analysis and number theory in interaction

Time-frequency analysis is a modern branch of Mathematics, with immediate applications in various fields such as signal and image processing, quantum physics, electrical engineering, seismic geology, radar, mobile communications, or the analysis of biometrical data. The basic object of time-frequency analysis is the Fourier transform and its time-variant form, the short-time Fourier transform. There is an intimate relation between the continuous-time point of view with its far developed theory, and the discrete-time point of view with its high importance for applications. For example, the discrete Fourier transform is a direct analogue of the continuous Fourier transform. Yet the discrete version opens new questions, which quickly lead to number theoretic problems that are not part of the continuous theory. We are especially interested in the interaction between these number theoretic problems and the analytic problems of the continuous-time theory. The benefit of this research will be a new understanding for the applicability of time-frequency analysis. The research will focus on the role of number theoretic questions for this part of analysis, but also on the use of Fourier and Time-frequency methods for Number theory.

This project was concerned with mathematical foundations in the interaction of two fields: time-frequency analysis and number theory. Problems and questions in this type of foundational research often come indirectly from applications. One of the central questions in time-frequency analysis is how to improve the transmission of data, especially wireless transmission (cell phones, WLAN, antenna TV). A key problem in wireless transmission is the reduction of interference. Interference means that one channel can disturb another channel, so that stable transmission can be maintained only at the cost of lower transmission rates. While interference can never be removed entirely, it can be reduced considerably by improved methods, which allows for higher transmission rates.Interference reduction is an engineering topic that is based on techniques with a mathematical background. An important method used today is interference alignment. The idea is that unwanted interferences from different sources are aligned, so that they need not be reduced one by one but instead can all together be reduced simultaneously in one step. The continuous improving of interference alignment is also effected by mathematical foundational research in engineering.Especially the results of metric Diophantine approximation are useful in this context, as pointed out by Adiceam et.al. in their article Diophantine approximation and applications in interference alignment, published 2016 in the important journal Advances in Mathematics. In a nutshell, metric Diophantine approximation describes how to approximate real numbers by rational numbers. The project has brought useful new contributions to this research field.