Mathematics

PIERO ANTONIO D'ANCONA
PIERO ANTONIO D'ANCONA

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

General objectives: To acquire basic notions of harmonic analysis related to the continuous and discrete Fourier transform and Fourier series, and to know the main applications of these methods to both theoretical and practical problems.

Specific objectives:

Knowledge and understanding: by the end of the course the student will have acquired the main notions about continuous and discrete Fourier transform, Fourier series, wavelets, and their use in some theoretical and practical fields (differential equations, image processing, signal theory).

Applying knowledge and understanding: at the end of the course the student will be able to solve basic level problems in harmonic analysis, will be familiar with Fourier transforms and Fourier series, and will be able to apply these techniques to the solution of various concrete problems.

Critical and Judgmental Skills: the student will have the basis to understand when harmonic analysis techniques can be useful as tools for solving problems in various fields of analysis and its applications.

Communication skills: ability to expose the contents in the oral part of the test and answer theoretical questions.

Learning ability: the acquired knowledge will allow a study, individually or in a course, of more advanced aspects of harmonic analysis, and of more specific applicative topics.

Fundamentals of Calculus for functions of one and several real variables, Lebesgue integral theory, elements of Functional Analysis and theory of Hilbert Spaces

Introduction to Fourier Analysis.
Fourier Transform on S, L^2 and S'
Fourier Transform on Lp
Methods of Harmonic Analysis
Explicit Calculation of Transforms
Uncertainty Principle
Fourier Series
Discrete Fourier Transform
Some applications:
- signal theory and Shannon-Nyquist Theorem
- representation of solutions of PDEs
- Strichartz estimates for the Schrodinger equation
- Weyl's equidistribution theorem
Multiresolution analysis: Haar and Daubechies wavelets

All course topics are contained in a series of notes prepared by the lecturer and available to students from the beginning of the course.
For more in-depth coverage of the topics covered in the course, see the additional bibliography

Fourier transform and Fourier series:
Katznelson: An Introduction to Harmonic Analysis
Dym, McKean: Fourier Series and Integrals
Stein, Shakarchi: Fourier Analysis

Discrete transforms, signal theory, wavelets:
Wong: Discrete Fourier Analysis
Nickolas: Wavelets, A Student Guide
Gasquet, Witomski: Fourier Analysis and Applications

Classroom and face-to-face lectures

Optional but strongly recommended

The final test consists of an oral examination on the results and methods described during the course

Discussion of one of the theoretical results studied during the course and application to concrete examples