Introduction to 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.

• 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

Lectures will be taught at the whiteboard, with an emphasis on building intuition and understanding of the theory step by step. While the approach is rigorous, the focus will be on clarity, motivation, and meaningful examples. Students are encouraged to ask questions and engage actively during the lectures. Theoretical results will be supported by examples from applied sciences to show how the abstract concepts come to life in practice. No prior familiarity with distribution theory is assumed, the necessary tools will be introduced gradually as needed. However, the fundamentals of mathematical analysis (differentiation, series, Riemann integrals) are strongly recommended.

(The modality may be discussed). Provisional modality: oral.