Distributionsteori

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

**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

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.

*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.

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.

Duistermaat, J.J. & Kolk, Johan A.C.: Distributions: Theory and Applications. BirkhĂ¤user, 2010. ISBN 9780817646721.

Available as e-book via the department library.

**Types of instruction:** Lectures, seminars

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

**Assumed prior knowledge:** MATC11 Analytic functions

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