250115 VO Frame Theory (2017W)

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

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik

Details

Sprache: Englisch

Prüfungstermine

Lehrende

Termine (iCal) - nächster Termin ist mit N markiert

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

Information

Ziele, Inhalte und Methode der Lehrveranstaltung

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.

Art der Leistungskontrolle und erlaubte Hilfsmittel

Written exam

Mindestanforderungen und Beurteilungsmaßstab

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.

Prüfungsstoff

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

Literatur

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

Zuordnung im Vorlesungsverzeichnis

MANV, MAMV

Letzte Änderung: Mo 07.09.2020 15:40