Polyanalytic functions and related topics

In this this project, I will work at the intersection of two major areas in mathematics, complex and time-frequency analysis. The former is the subfield where one studies functions defined on complex numbers. The main objects of interest there have traditionally been so called analytic functions, which can be geometrically described as mappings that map small circles to circles. In time- frequency analysis, one studies how a function can be represented in a way which shows its time-frequency contents in an accessible form. The field has its origins in applications and the central tool, short time Fourier transform, is very commonly used in engineering sciences. As an example of a useful time- frequency representation, one can think of the musical score. Indeed, there the signal (a piece of music in this case) is described in a way that tells which frequencies are present at any given moment. A unifying theme in this project is the theory of polyanalytic functions, a classical topic in complex analysis generalizing the notion of an analytic function, which provides a link between these two fields. One of the projects that I will work on deals with the following classical question in harmonic analysis: if the values of a function are measured at some given points, so called sampling nodes, can it be reconstructed from these values in a stable way? Because most real life signals are continuous whereas in practical applications a discrete version required, this type of question is important not only in mathematical analysis but also in engineering sciences. We will work on the theoretical background of this problem in variety of settings arising in complex and time-frequency analysis and try to give conditions on how dense the sampling nodes should to be for this stable recovery to be possible. I will also work on certain determinantal point processes related to complex and time-frequency analysis. A point process is by definition a random point configuration and determinantal point processes constitute an important subclass. These possess a very elegant mathematical theory and appear in many different areas such as statistical physics and number theory. The research in this project will have applications in the study of electrons in a strong magnetic field. I will also develop further the Uncertainty principle which in its original form says a signal cannot be too localized in time and frequency simultaneously. We will search for versions of this principle in several settings in time-frequency and complex analysis, with the aim of giving estimates of the concentration that are as sharp as possible.

This project centered around polyanalytic functions, which are classical objects in complex analysis - an area of mathematics that is specifically concerned about 2-dimensional phenomena. The general aim of the project was to investigate how polyanalytic functions act as a connecting link between some seemingly separate areas of mathematics. The main focus of the project was to study polyanalytic functions which are chosen randomly. Such random functions can appear naturally for example in signal analysis. In that field, it is a usual procedure to transform the signal of interest in such a way that it becomes easier to analyse. One of the most widely used transformations is the short-time Fourier transform. Applying the short-time Fourier transform to pure noise leads to the functions that were studied in this project. This very promising topic lies in the interface of quantum mechanics, time-frequency analysis, complex analysis and probability. I consider opening of this research line the most important contribution of the project. In addition, I investigated the following type of question, which appear in several areas of pure and applied mathematics: how can a full signal be recovered from measurements that give only partial partial information of that signal? I investigated this question in the context of complex analysis and succeeded with my collaborators in providing very accurate answers.