Lecture on Fourier Analysis (SS 2020)

Prof. Dr. Robert Denk

  • Time and place 15.15-16.45 in D 406 (as soon as regular lectures will be possible again).
  • Exercises every two weeks
  • Language: German

Because of the actual restrictions for lectures, this lecture will be provided in an online version. Therefore, it is important for all students to register in ILIAS to this lecture. In ILIAS, you will find details on the organization and the online contents.

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 are the first-year courses, some basic knowledge in functional analysis and in measure theory is useful. This lecture is classified as Wahlmodul for Bachelor and 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.

Exams:

Caused by the actual situation, the form of the final exams cannot be determined at the moment.