Prof. Dr. Robert Denk

  • Time: Monday, 13.30-15.00 in F 420
  • Exercise group (organized by Max Nendel), every second week. See the ILIAS system.
  • Language: German

Content
Literature
Lecture notes
Related Modules
Exam

Content:

The Fourier transform is a central topic in the field of analysis and typically occurs in the form of Fourier series or of the Fourier transform in  Rn. In this lecture, we will discuss these both variants and their properties. In the theory of Fourier series, one considers the representation of a periodic function as a series of sine and cosine terms (or the complex version of it). Typical questions are the convergence of the series and the reconstruction of the function. In the whole space case, the series becomes an integral. One example of a typical result is the theorem of Plancherel who states that the Fourier transform is an isometric isomorphism of the space  L2(Rn). Further topics to be discussed are the convolution of functions, Paley-Wiener theorems and the sampling theorem as well as the Fourier transform in the space of tempered distributions. 

Prerequisites for this lecture is(apart from the first-year courses) some basic knowledge in functional analysis and in measure theory. This lecture is intended for Bachelor students of the 5th semester and as Wahlmodul for master students.

 

Literature:

A list of references can be found at the end of the lecture notes.

Lecture notes:

There are lecture notes (in German) which will be updated regularly. The lecture notes can be found in ILIAS.

Related modules:

  • Bachelorstudiengang Mathematik (82105H)
      6500 Ergänzungsmodule
  • Masterstudiengang Mathematik (88105H)
      3000 Wahlmodule
         3100 Wahlmodule 
  • Lehramtsstudiengänge WPO 2001
         (Staatsexamen)
           Mathematik
             Hauptstudium Vorlesungen 
  • Diplomstudiengänge
         Mathematik
             Hauptstudium Vorlesungen

Exam:

To pass this module, one has to participate successfully in the exercise groups and to pass a final oral exam.