250115 VO Frame Theory (2017W)

(with Applications in Acoustics, Signal Processing and Quantum Mechanics)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik

Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

Tuesday 03.10. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 10.10. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 17.10. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 24.10. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 31.10. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 07.11. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 14.11. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 21.11. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 28.11. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 05.12. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 12.12. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 09.01. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 16.01. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 23.01. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock
Tuesday 30.01. 14:15 - 16:45 Seminarraum 10 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

Frame theory is concerned with the study of stable, potentially overcomplete spanning sets in a Hilbert space. Its starting point is a generalization of the principle of an orthonormal basis resulting in the definition of a frame. Similar to orthonormal bases (ONBs) every function can be (i) recovered from its frame coefficients, i.e. the inner products with respect to the frame elements and (ii) expanded into a linear combination of the frame elements. Frames have a rich structure despite being much less restrictive than ONBs, rendering them attractive for a wide number of applications. In addition to being an active field of research, posing interesting research questions of its own, frame theory has applications in other fields, like signal processing and physics.

Students of this course will gain understanding of the basic properties of frames and Riesz bases in comparison to ONBs, both in a linear algebra and functional anaylsis context. The implementation of frame-related algorithms will be considered and applications in acoustics, signal processing and quantum mechanics are presented.

For a short introduction see
https://en.wikipedia.org/wiki/Frame_(linear_algebra)

This will be a standard frontal course, using both blackboard and beamer.

Assessment and permitted materials

Written exam

Minimum requirements and assessment criteria

A basic understanding of concepts from functional analysis and linear algebra.

For a successful conclusion of this course, students must demonstrate knowledge of the basic concepts and theorems, as well as an understanding of the main proofs and applications presented.

Examination topics

Everything that is covered in the course, i.e.
1.) Spanning sets in finite dimensional vector spaces
2.) Bessel sequences
3.) Riesz bases
4.) Frames
5.) Generalized shift-invariant systems

Reading list

Ole Christensen, An Introduction to Frames and Riesz Bases (among others)

Association in the course directory

MANV, MAMV

Last modified: Mo 07.09.2020 15:40