Transform Theory, 4 credits

Main field of study

Course level

Advancement level

Course offered for

• Industrial Engineering and Management - International, M Sc in Engineering
• Biomedical Engineering, M Sc in Engineering
• Applied Physics and Electrical Engineering - International, M Sc in Engineering
• Physics, Bachelor´s Programme

Entry requirements

Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.

Prerequisites

Intended learning outcomes

The course aims to give students a deeper knowledge of Fourier analysis and Transform Theory , which have many applications in both technology and mathematics. After successfully completing the course the student is expected to

• be acquainted with necessary conditions for the existence of the transforms,
• know and be able to derive simple properties of the transforms (e.g. behaviour at infinity, scaling and translation rules, rules for differentiation and integration as well as rules for multiplication by the time variable).
• be able to derive the transforms of the elementary funktions,
• know the invversion theorems, uniqueness theorems, the convolution formulas and the Parseval and Plancherel theorems,
• be able to use the transforms to solve problems such as differential equations, Difference equations and convolution equations
• be acquainted with and be able to use results about uniform convergence (continuity, differentiability and integrability of limit functions, Weierstrass’ Majorant Theorem).

Course content

In this course we study some important linear transformations which allow us to translate linear problems (differential, integral and difference equations) into more tractable algebraic problems, whose solutions can then be translated back to solutions of the original problem.
We study: Fourier series, which translate periodic functions into function series. These series are used to analyze periodic behaviour. The problem of convergence of the function series is important and we look att uniform and pointwise convergence as well as convergence in the mean for Fourier series. Bessel’s and Parseval’s Theorems are key results. Fourier transforms: these transforms are used to analyze non-periodic behaviour. The inversion formula for Fourier transforms is of central importance, and other tools at our disposal include the rules of calculation, the convolution formula and Plancherel’s Theorem. The Laplace transform: this transforms functions of a real variable into functions defined in the complex plane and it is used amongst other things for solving initial value problems. The tools at our disposal include rules of calculation, the convolution formula as well as initial and final value theorems. The Z-transform: transforms functions of the natural numbers into power series, and it is used to solve difference equations. The tools at our disposal include rules of calculation and the convolution formula.

Teaching and working methods

Teaching is done through lectures and problem classes.

Examination

Grades

Department

Director of Studies or equivalent

Examiner

Education components

Course literature