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250005 VO Time-Frequency Analysis (2021S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik

Moodle

Registration/Deregistration

Note: The time of your registration within the registration period has no effect on the allocation of places (no first come, first served).

Details

Language: English

Examination dates

Lecturers

Markus Faulhuber

Classes (iCal) - next class is marked with N

According to the current regulations, the course will probably be held in digital form. To produce an atmosphere which is close to a mathematical lecture course, the topics will be presented "live" via a digital black board. Further information and links will be provided in moodle.

Tuesday 02.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 04.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 09.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 11.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 16.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 18.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 23.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 25.03. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 13.04. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 15.04. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 20.04. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 22.04. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 27.04. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 29.04. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 04.05. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 06.05. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 11.05. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 18.05. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 20.05. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 27.05. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 01.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 08.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 10.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 15.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 17.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 22.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Thursday 24.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 29.06. 11:30 - 13:00 Digital
Seminarraum 12 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

This lecture course is an introduction to the theory of time-frequency analysis and related topics.

After a short recap on basic results from functional analysis and Fourier analysis, we will introduce time-frequency shifts and the short-time Fourier transform (STFT). These are the central tools in this course. Nowadays, these concepts, along with related concepts such as the wavelet transform, form the basis of modern signal processing and data transmission (e.g. WLAN, 4G/5G).

After introducing the STFT, we will study related "quadratic representations" of a function and their connections to the STFT.

Furthermore, we will study the Poisson summation formula, which is an essential tool in time-frequency analysis, but also in mathematical physics (energy minimization and lattice structures).

Motivated by finding an "ideal time-frequency representation", we will encounter the classical Heisenberg-Pauli-Weyl uncertainty principle. It appears in a natural way as a consequence of the "canonical commutation relations", which are intimately connected to the commutation relations of time-frequency shifts. Based on that, we will get to know further uncertainty principles, obeying 3 "meta principles".

After the study of these fundamental principles, we will introduce the concept of frames, which are a generalization of bases. In particular, we will study Gabor systems and Gabor frames.

In the remaining part of the lecture course, we will deal with more abstract topics. These may include:
- the symplectic and the metaplectic group
- representation theory of the Heisenberg group
- relevant function spaces, in particular modulation spaces (i.a. Feichtinger's algebra) and Wiener amalgam spaces
- the Zak transform
- duality theory for Gabor systems
- the Balian-Low theorem and density results

To some extend the presented results are part of the current state-of-the-art of science. Some related open problems might as well be discussed.

Assessment and permitted materials

Depending on the number of participants, the exam will be either oral or written. The number of participants will be evaluated during the first few lectures.

Minimum requirements and assessment criteria

Positive examination.

Examination topics

Material presented in the lecture course.

Reading list

- Gerald B. Folland. Harmonic analysis in phase space. Princeton University Press, 1989
- Maurice A. de Gosson. Symplectic Methods in Harmonic Analysis and in Mathematical Physics. Birkhäuser/Springer Basel AG, 2011
- Karlheinz Gröchenig. Foundations of Time-Frequency Analysis. Birkhäuser, Boston, MA, 2001

Lecture notes will be available as well.

Association in the course directory

MANV; MAMV;

Last modified: Fr 12.05.2023 00:21