Continuous and discrete Gabor frames

Time-frequency analysis is a flourishing mathematical topic. Time-frequency analytic techniques are successfully applied in various fields, such as quantum physics, electrical engineering, seismic geology, radar, mobile communications, or the analysis of biometrical data. Because of their interdisciplinary origin, these techniques appeal to scientists and engineers of many different backgrounds. The typical functional analytic model in time- frequency analysis are Gabor frames in a continuous-time setting. On the other hand in applications the attention is often put on discrete models, computational methods, and numerical results. We establish a link between the continuous and the discrete setting. The use of the discrete model is threefold. First, there are important approximation results that describe a continuous Gabor frame as a limit of discrete Gabor frames. Thus the discrete model has direct implications for the continuous theory. Secondly, the discrete and continuous theory often leads to analogous problems. Thus even in absence of direct approximate implications the discrete analysis contributes to understanding the continuous problem by analogy. Thirdly, as indicated above, discrete Gabor frames are highly relevant for applications and thus the discrete setting is an area of interest in its own right. In fact, we will also describe discrete phenomena that do not have an obvious continuous analogue.

The classic theory of Fourier analysis is concerned with the decomposition of functions or signals into basic waveforms, the sine and cosine functions. In the original form, a Fourier series decomposes the finite signal of a vibrating string into integer multiples of a pure frequency, the harmonics. A continuous-time analogue is the Fourier integral, the decomposition of an infinite signal into arbitrary multiples of a pure frequency. Yet often one is not only interested in the frequency content of the whole signal, but in the frequency content of portions of the signal, supporting the idea that these frequencies may vary over time. A powerful tool for this type of analysis are time-frequency analytic methods. The standard operator in time-frequency analysis is the short-time Fourier transform. The shorttime Fourier transform computes, for each moment in time, the frequency content of the signal near that point of time. We notice that for the concept of a frequency content at a point of time, there are certain limitations, mathematically well understood and expressed in general forms of the Heisenberg uncertainty principle. These limitations are typically overcome by a careful, application-oriented choice of the so-called window function for the short-time Fourier transform. For example, a standard window-function is the Gaussian bell function, whose width (standard deviation) can be adapted to the signal under investigation. The window function is shifted along the infinite signal and its role is to localize the signal, essentially extracting a finite part of it. The result of this approach is a powerful, flexible tool for the analysis of signals, with immediate extensions to more-dimensional applications such as image analysis. Indeed time-frequency analysis is a mathematical field with applications in various sciences and technologies, such as electrical engineering, wired and wireless communications, seismology, the analysis of geological structures, audio processing for hearing aids, speech recognition, the analysis of biometrical data and heart rhythm monitoring. Usually, in these applications, the data is given in an analogue form, which is the common way in which we understand data that is found in nature. But computer software is mostly designed to treat data in digital form, the common way that we put data into computers. Some mathematical fields focus either on the analogue side or on the digital side. Time-frequency in fact provides us with a theory dealing with analogue objects (functions) as well as a separate theory for dealing with discrete objects (vectors). The project is based on the connection between the analogue theory (continuous-time timefrequency analysis) and the discrete theory (discrete-time time-frequency analysis). In particular the project has developed new mathematical results in the discrete theory that indeed represent methods from the analogue theory. Also, the project was possible to provide fundamental new insight into the analogue theory, especially with a well-known time-frequency analytic phenomenon called the Balian-Low Theorem. Pursuing the project also led to the use and investigation of number theoretic problems. Many deep relations between Fourier analysis and number theory are known, and the project contributes to intensifying this connection.