From Linear Algebra to Fourier Analysis - NMAG359

• Title: From Linear Algebra to Fourier Analysis
• Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2019
• Semester: winter
• E-Credits: 3
• Hours per week, examination: winter s.:2/0, Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: not taught
• Language: English
• Teaching methods: full-time
• Teaching methods: full-time
• Guarantor: prof. Hans Georg Feichtinger
• Class: M Bc. MMIB > Povinně volitelné; M Bc. OM > Povinně volitelné

Annotation

Last update: doc. Mgr. Petr Kaplický, Ph.D. (01.06.2018)

An exceptional elective course (povinně volitelný) intended for lectures that takes place just once.

Literature

Last update: doc. Mgr. Petr Kaplický, Ph.D. (13.06.2018)

1. E. Cordero, H. G. Feichtinger, and F. Luef. Banach Gelfand triples for Gabor analysis. In Pseudo-differential Operators, volume 1949 of Lecture Notes in Mathematics, pages 1-33. Springer, Berlin, 2008.
2. H. G. Feichtinger and T. Strohmer. Gabor Analysis and Algorithms. Theory and Applications. Birkhäuser, Boston, 1998.
3. H. G. Feichtinger and T. Strohmer. Advances in Gabor Analysis. Birkhäuser, Basel, 2003.
4. G. B. Folland. Harmonic Analysis in Phase Space. Princeton University Press, Princeton, N.J., 1989.
5. K. Gröchenig. Foundations of Time-Frequency Analysis. Appl. Numer. Harmon. Anal. Birkhäuser, Boston, MA, 2001.

For the courses a dedicated web-page will be installed, in order to facilitate communication between the course leader and the participating students, comparable to www.nuhag.eu/TUM (used in 2017 and 2018). Further literature is found via www.nuhag.eu/bibtex, see [1-3]. MATLAB code can be obtained from here .

Syllabus

Last update: doc. Mgr. Petr Kaplický, Ph.D. (13.06.2018)

The goal of the course will be to introduce all the key concepts of time frequency analysis, in particular of what is called Gabor Analysis (a kind of discretization of the time-frequency plane, also called phase-space).

The starting point will be linear algebra and polynomials as an important special case. The participants will gain access to the NuHAG based MATLAB toolbox and also learn about the key features of the so-called LTFAT toolbox (Large Time-Frequency Toolbox) hosted by ARI (Acoustic Research Institute, Vienna).

Question of time-frequency analysis over finite Abelian groups are related to problems in digital signal processing. For the one-dimensional we are concerned with audio signals, for the two-dimensional case it is about digital image analysis. The course will lead up to ideas about time-variant (resp. space-variant) filtering or the principles of mobile communication, viewed from a linear algebra point-of-view.

Entry requirements

Last update: doc. Mgr. Petr Kaplický, Ph.D. (01.06.2018)

This course will require from the participating students only a general background in linear algebra, and knowledge of the Riemann integral, ideally with some interest in numerical computations (if not, the examples which will be presented can be viewed just as illustrations of concrete linear algebra applications, providing concrete numerical results, with some discussion of numerical efficiency). In this sense the course should be accessible to students after the first year of studies, ideally in the third year of studies and further on, but even advanced (say PhD) students will benefit from the un-orthodox view on linear algebra concepts, as realized using the programming language MATLAB, which is used widely by applied mathematicians, engineers or physicists. Previous knowledge of MATLAB will not be required, although certainly helpful. Also, following the course without the desire to learn or use MATLAB is also possible.