Universität Wien
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250103 VO Frame Theory (2023W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik
ON-SITE

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

First meeting 03.10.2023 16:45

  • Tuesday 03.10. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 04.10. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 10.10. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 11.10. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 17.10. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 18.10. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 24.10. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 25.10. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 31.10. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Tuesday 07.11. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 08.11. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 14.11. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 15.11. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 21.11. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 22.11. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 28.11. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 29.11. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 05.12. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 06.12. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 12.12. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 13.12. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 09.01. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 10.01. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 16.01. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 17.01. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 23.01. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 24.01. 16:45 - 17:30 Seminarraum 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 30.01. 16:45 - 18:15 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Wednesday 31.01. 16:45 - 17:30 Seminarraum 11 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, quantum mechanics and machine learning will be presented as motivation.

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

This will be a standard frontal course, using mostly the blackboard and ocaasionally the beamer.

Assessment and permitted materials

Written or oral exam.

Minimum requirements and assessment criteria

A basic understanding of concepts from functional analysis and linear algebra is expected for students to follow the course.

For a successful conclusion of this course, students must demonstrate knowledge of the basic concepts and theorems, a basic understanding of the main proofs and applications presented, as well as a ability to use the techniques in similar results.

Examination topics

Everything that is covered in the course, and presented.

The current plan is:
1.) Spanning sets in finite dimensional vector spaces
2.) Bessel sequences
3.) Riesz bases
4.) Frames
5.) Special topic in frame theory: e,g, phase retrieval, localization, ...

Reading list

The course will mostly stick to
Ole Christensen, An Introduction to Frames and Riesz Bases

Association in the course directory

MAMV; MANV

Last modified: Mo 08.07.2024 09:46