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 compact Lie

groups and on compact 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

to the monograph

M. Ruzhansky, V. Turunen

Pseudo-Differential Operators and Symmetries

Birkhäuser 2010.