Course syllabus - Fourier Analysis

Objectives

The objective of the course is to give the student opportunity to acquire basic understanding of the theory of orthogonal series and transforms, with applications in engineering and physics.

Learning outcomes

Upon completion of the course, the student is expected to be able to

1. apply the Laplace transform to find solutions to linear differential and integral equations with given initial conditions
2. determine pointwise convergence of Fourier series, and explain the Gibbs phenomenon
3. determine orthogonality and completeness of systems of functions in L2, and find optimal approximations of L2 functions with respect to linear subspaces spanned by orthogonal systems
4. calculate eigenvalues and eigenvectors/eigenfunctions, with and without a computer
5. solve boundary value and initial value problems, including the one-dimensional heat equation, the one-dimensional wave equation, and the Laplace equation in two dimensions
6. apply the Fourier transform to functions on unbounded intervals, including the application of the inversion theorem and Plancherel's formula

Course content

- Laplace transform, Fourier transform
- Fourier series, eigenfunctions and eigenvalues
- Fourier methods and separation of variables
- Sturm-Liouville problems, L2-theory, orthogonal systems
- Basic distributions, the Heaviside function, Dirac's delta
- Examples of applications of Fourier analysis in e.g. physics and signal processing

Specific requirements

45 credits in Mathematics/Applied Mathematics which includes Single Variable Calculus, 5 credits, Calculus of Several Variables, 5 credits, and Vector Algebra, 5 credits, or equivalent.

Examination

LAB1, Laboratory work, 1.5 credits, computer lab with written and oral report concerning learning outcome 4, grades Fail (U) or Pass (G).
INL1, Written assignment, 3 credits, written assignment concerning learning outcomes 1-6, grades Fail (U), 3, 4 or 5.
TEN1, Written examination, 3 credits, individual written exam concerning learning outcomes 1-6, grades Fail (U), 3, 4 or 5.
 
For grade 4 on the course as a whole, the student must have earned a grade average of 4 for INL1 and TEN1. For grade 5 on the course as a whole, the student must have earned that grade for INL1 and TEN1.

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Grading scale: 5, 4, 3