Fourier Analysis
A.Y. 2024/2025
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
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 mandatory prerequisit. However, a good knowledge of (many of) the topics of the course Analisi Reale is strongly suggested.
Teaching methods
Classroom lessons with the use of a blackboard. Lecture notes provided.
Teaching Resources
-G. Folland, Real Analysis
-L. Grafakos, Classical Fourier Analysis
-Y. Katznelson, An Introduction to Harmonic Analysis
-M. M. Peloso, Appunti del corso
-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.
- 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
Professor:
Salvatori Maura Elisabetta
Shifts:
Turno
Professor:
Salvatori Maura ElisabettaEducational website(s)
Professor(s)