Time-frequency analysis and pseudo-differential operators

Time-frequency analysis is a branch of mathematics stemming from traditional Fourier analysis. Motivating applications come from signal processing (e.g. denoising and  compressing sounds and pictures), and from partial differential equations: wave propagation (hyperbolic), diffusion (parabolic), steady-state phenomena (elliptic). Time-frequency methods have been used in quantum mechanics, medical imaging, radar detection, speech and music analysis, telecommunications, geophysics, atmospheric physics etc. In practical situations, one needs efficient computational methods, such as Fast Fourier Transform.

Time-frequency problems arise also from abstract mathematical considerations. Yet, theory and applications are closely related to each other here. Often there is an underlying symmetry group that in fact gives rise to a global Fourier transform on the space. This leads to studying representation theory and harmonic analysis on groups, and related geometry.

One of the main objectives in the time-frequency analysis is to decompose a signal in a way that shows its energy content jointly in both time and frequency. After such decomposition, we may operate on the signal in a controlled way, and construct signals that have specific properties. The fundamental basic tool is the classical Fourier transform, which is most suitable for time-stationary signals. For signals evolving in time, we may use linear time-frequency representations (such as short-time Fourier transforms, wavelet transforms etc.), or quadratic time-frequency representations (for instance Cohen’s class, including the Wigner distribution originating from quantum mechanics). Each of these methods has some strengths and weaknesses.

A signal manipulation (e.g. denoising) can be presented as a pseudo-differential operator, which can be thought as a weighted inverse Fourier transform. Pseudo-differential operators were originally introduced in the context of elliptic partial differential equations. Such operators appear naturally when reducing elliptic boundary value problems to the boundary.

We study pseudo-differential operators globally on Lie groups and on symmetric spaces, without resorting to local charts. This can be done by presenting functions on the symmetry group by Fourier series coming from the representations of the group. Consequently, global calculus and full symbols of pseudo-differential operators are obtained. For students and experts alike, for more information on related analysis, we refer to the monographs on compact and nilpotent Lie groups, respectively:

M. Ruzhansky, V. Turunen. Pseudo-Differential Operators and Symmetries. Birkhäuser, 2010. extracts

V. Fischer, M. Ruzhansky, Quantization on Nilpotent Lie Groups. Progress in Math., vol. 314, Birkhäuser, 2016. download