# FOURIER ANALYSIS

OVERVIEW

This course presents an introduction to Fourier Analysis.

## AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide an introduction to the ideas and the methods of Fourier analysis, on the torus and on the real line.

AIMS AND LEARNING OUTCOMES

**Aims**

The aim of this course is to teach some classical topics in Fourier Analysis, which are considered fundamental for the preparation of the students of the Master's degree in Mathematics.

**Expected learning outcomes**

At the end of the course, the student will have to know the theoretical concepts introduced in the lectures, construct and discuss examples related to each of them (in such a way to better understand the abstract concepts), write/reconstruct the proofs seen in the lectures or easy variants of those and solve problems on the topics of the course.

PREREQUISITES

The basic concepts of functional analysis and measure theory (Elements of Advanced Analysis 1).

Teaching methods

The course consists of frontal lectures carried out by the teacher where the theory is explained and where basic examples are discussed (four hours per week). These are integrated with problem lectures (two hours per week): the students will prepare some exercises and discuss them with the teacher and the other students.

SYLLABUS/CONTENT

**Fourier series.** The space of periodic square summable functions. Orthonormal bases. Fourier series. Gibbs phenomenon. Fourier transform of periodic absolute integrable functions. Applications: spectral methods for partial differential equations.

**Fourier integrals. **Fourier integral of absolute integrable functions on R^n. Fourier transform of elementary fiunzions. Convolution. Approximate identities. Inversion formula. Fourier transform of square integrable functions. Poisson summation formula. The Paley-Wiener theorem. Shannon theorem.

**Tempered distributions.** The Fourier transform of tempered distributions.

RECOMMENDED READING/BIBLIOGRAPHY

V. Del Prete, Introduzione all'analisi di Fourier, Lecture notes (aulaweb)

Y. Katznelson, An introduction to harmonic analysis, Collocaz Bibl. DIMA 43-1968-07.

E. O. Brigham, The Fast Fourier Transform, Prentice Hall Englewood Cliffs, Boston,1974. -

H. Dym - H. P. Mc Kean, Fourier Series and Integrals, Academic Press, 1972.

I. Korner, Fourier Analysis, 1995.
- I. Korner, Exercises for Fourier Analysis, 1995.

E. Prestini, Applicazioni dell'analisi armonica, U.Hoepli, Milano, 1996I.

E. Prestini, The Evolution of Applied Harmonic Analysis. Models of the Real World Series, A Birkhäuser 2004.

G.B. Folland, Fourier analysis and its applications, Collocaz Bibl. DIMA 42-1992-01.

## TEACHERS AND EXAM BOARD

**Ricevimento:** By appointment

## LESSONS

Teaching methods

The course consists of frontal lectures carried out by the teacher where the theory is explained and where basic examples are discussed (four hours per week). These are integrated with problem lectures (two hours per week): the students will prepare some exercises and discuss them with the teacher and the other students.

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

The exam consists in an oral test.

Assessment methods

During the oral exam, the theoretical results and some problems are discussed. This allows to test the knowledge of the theory of the students and their abilities to put it into practice.