Fourier Analysis

Fourier analysis lies at the intersection of harmonic and microlocal analysis, engineering, signal processing, number theory, and compressed sensing, among other fields. This course aims to introduce Fourier transforms in various contexts, leading to the formulation of sampling theory and the development of modern tools for signal analysis. The course begins with an informal introduction to essential topics in modern analysis, fundamental to understanding Fourier analysis. Next, the Fourier transform for both periodic and non-periodic functions will be introduced. Uncertainty principles and sampling theory will be discussed at the end of the course.

Disclaimer: The prerequisite Probability & Measure has to be considered highly suggested, yet not really necessary.

• Provide the attendee of the fundamentals of modern mathematical analysis in a compact and accessible form (Hilbert spaces, Lp spaces, tempered distributions).
• Understand the definition and and the construction of Fourier series and the Fourier transform, and their main properties.
• State central results such as the Riemann-Lebesgue lemma, Parseval's identity, and the convolution and the differentiation theorems (with an idea of applications to partial differential equations such as Schrödinger equation and heat equation).
• Understand the relation between smoothness classes and Fourier transform decay.
• Understand the rationale behind uncertainty principles (Heisenberg's UP, Hardy's UP, and so on) and Shannon's sampling formula from a mathematical perspective.

In presenza

Frontal lectures

The exam will be oral.