Frequency Analysis and Applications

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).