Fourier Analysis

A.Y. 2020/2021
Overall hours
Learning objectives
The course gives the basis about the classical theory on the Fourier series and the Fourier transform both in the 1-dimensional case and in several dimensions.
Expected learning outcomes
Learning the basics facts about the convergence and the summability of Fourier series; properties of the Fourier transform when defined on principal function spaces and on distributions.
Course syllabus and organization

Single session

Lesson period
First semester
All lectures will be videolessons obtained and recorded using Microsoft Teams or Zoom. Recorded files will also be posted on Ariel. Both the program and the reference material do not change. The oral examination will be in presence or not following the University's instructions.
Course syllabus
Fourier series in one dimension. Principal properties of Fourier coefficients. Fejer and Dirichlet kernels, summability in norm and pointwise summability. Fourier transform in R and in R^n. Theory L^1 and L^2. Spaces of Schwartz functions S and of tempered distributions S' and Fourier transforms in S and S'. L^p theory. Hilbert trasform and singular integrals. Fourier multipliers and boundedness in L^p. Fourier series in more dimensions and norm convergence in L^p. Poisson's summation formula, Paley-Wiener theorems and Shannon's sampling theorem.
Prerequisites for admission
There are no prerequisites. Knowledge of many of the topics covered in the Real Analysis course may be helpful.
Teaching methods
Frontal lessons with the use of the blackboard (chalk or white).
Teaching Resources
-G. Folland, Real Analysis
-L. Grafakos, Classical Fourier Analysis
-Y. Katznelson, An Introduction to Harmonic Analysis
-M. M. Peloso, Appunti del corso
Assessment methods and Criteria
The final examination consists of an oral exam.
- In the oral exam, the student will be required to illustrate concepts, examples and results presented during the course and will be required to solve problems quite similar at those presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Educational website(s)