Spanning properties of Gabor systems

Given a square-integrable function on the real line, which will be referred to as the window, a countable set of time-frequency translates of this function is called a Gabor system. The most standard example of a window is the Gaussian. It is a fundamental question in pure mathematics and engineering sciences to determine spanning properties of such systems in terms of the window and the countable set in the plane which defines the step sizes of the time-frequency translates. In particular, it is important to decide whether such a system constitutes a frame. This means that any square-integrable function can be written as a superposition of functions of the system in such a way that the dependence between the function and the coefficients is stable. A description of Gabor frames is an open question in this generality. The aim of my work under the fellowship period will be to extend the present understanding of Gabor systems to a larger class of windows. In the research plan, I will describe two research directions in more detail. The first one is related to vector-valued Gabor systems generated by a vector of first Hermite functions. This vector is a natural analogue of the Gaussian in the scalar-valued case. The aim here is to extend previous results characterizing frames to cover non-uniform time- frequency translates. This question is equivalent to a sampling problem in spaces of polyanalytic functions which are square-integrable with respect to a Gaussian weight in the complex plane. The other topic is to understand completeness of scalar-valued Gabor systems when the step size of translation is constant in both time and frequency. The plan is to show that such Gabor systems are always complete given that the window is analytic and exponentially decaying and that the density of the set describing the step sizes of the time-frequency translates is greater or equal to 1. It is known that such general conditions do not guarantee that the system is a frame. It is hoped that studying completeness will also give new insights about the frame property.

The research carried out in this project concerned certain topics in two major areas in mathematics, complex and time-frequency analysis.In time-frequency analysis, the goal is to understand how one can represent signals in time and frequency simultaneously. A typical example of such a representation is a musical score, where the key feature is that the signal (a piece of music in this case) is described in a way that tells which frequencies are present at any given moment. This way of analyzing signals has turned out to be extremely useful in many applications. For more general signals, there are many different ways how such a representation can be made and it is an important topic to understand the differences between these possible approaches. After choosing an adequate way of viewing the signal in the time-frequency plane, the following question naturally arises: is it possible to find a discrete set of measurements so that the signal is uniquely determined by these measurements. In such case, all the information about the signal is carried in these measurements. This type of results are very important e.g. in telecommunications where it is only possible to transmit discrete information. One of the results in this project gave a condition for a discrete set in the time-frequency plane which guarantee that this is possible and thus provides theoretical background for methods used in engineering applications.Another topic concerned statistical physics of electrons in the plane when subjected to a strong perpendicular magnetic field. Such models can be analyzed using complex analytic methods. It is known that in this model, the electrons tend to occupy a certain equilibrium configuration. There are however certain fluctuations around this equilibrium. The results obtained in this project precisely describe these fluctuations.The third topic was about the uncertainty principle in quantum mechanics. In its classical form, this result says that one cannot know the position and the momentum of a particle precisely. We were able to obtain new sharp forms of this principle and also apply it to obtain results about Dirichlet series, classical mathematical objects arising e.g. in the study of prime numbers.