Abstract:
The goal of this thesis is to study signals that have a regularity property defined in
the frequency space, such as a decay on average of the amplitude of their Fourier
transform, by using techniques from frequency analysis. Frequency analysis is a set
of techniques that involve an analysis in the Fourier domain. We review some of
these techniques and some principles. More precisely we will decompose a signal into
countable sums of functions of which the Fourier transform is compactly supported in
a ball or an annulus by performing a Littlewood–Paley decomposition. We will apply
this technique to study the properties of functions having a specific regularity. Over
two hundred years ago, Fourier studied problems related to the series expansions
of periodic signals using elementary trigonometric polynomials. The theory was
extended to non-periodic signals by using the Fourier transform and forms the core
of harmonic analysis. Harmonic analysis is used in various fields such as signal
processing and partial differential equations (PDEs).