Fourier Analysis

5 credits

Syllabus, Bachelor's level, 1MA035

This course has been discontinued.

A revised version of the syllabus is available.

Code: 1MA035
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G2F
Grading system: Pass with distinction (5), Pass with credit (4), Pass (3), Fail (U)
Finalised by: The Faculty Board of Science and Technology, 15 March 2007
Responsible department: Department of Mathematics

What is a course syllabus?

Entry requirements

Transform Methods, Linear Algebra II

Learning outcomes

In order to pass the course (grade 3) the student should

understand the distinction between pointwise and uniform convergence and be able to decide whether a given sequence of functions is uniformly convergent;

know the properties of the Fejèr and Dirichlet kernels;

know some sufficient conditions for pointwise and uniform convergence of Fourier series;

be able to give an account of the theory of complete orthogonal systems and know how to apply it to Fourier series;

be familiar with the rules for Fourier transformation and know how to apply them;

be familiar with the L²-theory for the Fourier transform;

know how to define and apply the fast Fourier transform;

be able to formulate important results and theorems covered by the course and to describe the main features of the proofs of important theorems;

be able to use the theory, methods and techniques of the course to solve mathematical problems;

be able to present mathematical arguments to others.

Content

Series of functions, uniform convergence. Fourier series: convergence theorems and L²-theory. Orthogonal systems. The Fourier transform. Applications to ordinary and partial differential equations. The discrete Fourier transform, the fast Fourier transform.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.