Distribution Theory
Distributionsteori

MATP11F, 7.5 credits

Valid from: Autumn 2019
Decided by: Professor Thomas Johansson
Date of establishment: 2019-10-08

General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: MATP11
Teaching language: English

Aim

The theory of distributions makes it possible to, in a consistent way, extend the definitions of classic concepts in mathematical analysis, such as derivatives, integrals and Fourier transforms, to more general functions. The aim of the course is to give the PhD student solid knowledge in basic distribution theory in order to facilitate future research in, e.g., the theory of partial differential equations or control theory.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to account for the concept of a test function and its importance in the theory of distributions.
• be able to account for the definition of a distribution, and for the basic operations on distributions, such as differentiation, convergence of sequences and multiplication with smooth functions
• be able to explain how the Fourier transform of distributions is defined,
• be able to explain some basic concepts in the theory of linear partial differential equations, such as hypoelliptic operator and fundamental solution
• be able to account for the Schwartz kernel theorem and the corresponding result for translation invariant linear maps.
• .

Competences and Skills

For a passing grade the doctoral student must

• in simple cases be able to determine whether a given function corresponds to a distribution
• be able to determine the Fourier transform of tempered distributions
• be able to solve, in typical cases, linear partial differential equations with constant coefficients using fundamental solutions.

Judgement and Approach

For a passing grade the doctoral student must be able to elaborate on the difference between solving problems in the sense of distributions and solving them in the classical sense.

Course Contents

The foundations of distribution theory. Test functions, the concept of distributions, distributions with compact support, operations on distributions, convolution, homogeneous distributions and the Fourier transform of a tempered distribution.

Course Literature

Duistermaat, J.J. & Kolk, Johan A.C.: Distributions: Theory and Applications. Birkhäuser, 2010. ISBN 9780817646721.
Available as e-book via the department library.

Instruction Details

Types of instruction: Lectures, seminars

Examination Details

Examination formats: Written exam, oral exam
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: MATC11 Analytic functions

Course Occasion Information

Contact and Other Information

Course coordinators:
Web page: http://www.ctr.maths.lu.se/course/disttheo/