Distribution Theory
Distributionsteori

MATP11F, 7.5 högskolepoäng

Gäller från och med: Autumn 2019
Beslutad av: Professor Thomas Johansson
Datum för fastställande: 2019-10-08

Allmänna uppgifter

Avdelning: Mathematics
Kurstyp: Gemensam kurs, avancerad nivå och forskarnivå
Kursen ges även på avancerad nivå med kurskod: MATP11
Undervisningsspråk: English

Syfte

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.

Mål

Kunskap och förståelse

För godkänd kurs skall doktoranden

• 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.
• .

Färdighet och förmåga

För godkänd kurs skall doktoranden

• 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.

Värderingsförmåga och förhållningssätt

För godkänd kurs skall doktoranden be able to elaborate on the difference between solving problems in the sense of distributions and solving them in the classical sense.

Kursinnehåll

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.

Kurslitteratur

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

Kursens undervisningsformer

Undervisningsformer: Föreläsningar, seminarier

Kursens examination

Examinationsformer: Skriftlig tentamen, muntlig tentamen
Betygsskala: Underkänd, godkänd
Examinator:

Antagningsuppgifter

Förutsatta förkunskaper: MATC11 Analytic functions

Kurstillfällesinformation

Kontaktinformation och övrigt

Kursansvariga:
Hemsida: http://www.ctr.maths.lu.se/course/disttheo/