[AN] Ivan Trapasso: Explorations in wave packet analysis
In this talk we provide a concise overview of the fundamental principles underlying harmonic analysis in phase space. The roots of this vibrant field of modern Fourier analysis are to be found at the crossroads of signal analysis, mathematical physics, representation theory and analysis of partial differential equations. The key idea is to exploit a dictionary of oscillating wave packets (or equivalently, the combined structure of translations and modulations or dilations) to investigate properties of functions, distributions and operators in terms of suitable companion phase space representations.
Addressing time and frequency/scale on the same level presents both advantages and challenges due to the uncertainty principle. In essence, time and frequency exhibit a somewhat dual nature as variables, hence the efforts to handle them concurrently are ultimately directed to keep track of the multifaceted manifestations of their entanglement. We will delve into these issues, whose origins date back to the foundations of quantum mechanics, and show how they continue to stimulate insightful research in analysis.
Lastly, we will offer a taste of applications of these techniques to some problems motivated by the current challenges of data science, mostly in order to convey the message that the principles of time-frequency analysis are ubiquitous, hence adopting a phase space perspective can provide a versatile framework to explore problems from pure and applied mathematics.
