Third-Cycle Courses

Faculty of Engineering | Lund University

Details for the Course Syllabus for Course MATP11F valid from Autumn 2019

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  • 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.
  • 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.
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.
Types of Instruction
  • Lectures
  • Seminars
Examination Formats
  • Written exam
  • Oral exam
  • Failed, pass
Admission Requirements
Assumed Prior Knowledge
Selection Criteria
  • Duistermaat, J.J. & Kolk, Johan A.C.: Distributions: Theory and Applications. Birkhäuser, 2010. ISBN 9780817646721.
  • Available as e-book via the department library.
Further Information
Course code
  • MATP11F
Administrative Information
  • 2019-10-08
  • Professor Thomas Johansson

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