Universität Wien

250071 VO Frame Theory (2025W)

5.00 ECTS (3.00 SWS), SPL 25 - Mathematik
Tu 16.12. 16:45-17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock

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Details

Language: English

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 01.10. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 07.10. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.10. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 14.10. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.10. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 21.10. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 22.10. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 28.10. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.10. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 04.11. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 05.11. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 11.11. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 12.11. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 18.11. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.11. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 25.11. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.11. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 02.12. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 03.12. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 09.12. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 10.12. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 17.12. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 07.01. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 13.01. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 14.01. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 20.01. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 21.01. 16:45 - 18:15 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 27.01. 16:45 - 17:30 Seminarraum 8 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 28.01. 16:45 - 18:15 Seminarraum 7 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. Applications in acoustics, signal processing and quantum mechanics are 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

Depending on the number of students written or oral 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.) Particular frame systems: Gabor, Wavelets, Shift-invariant Systems

Reading list

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

Association in the course directory

MAMV; MANV; ML2; MEL

Last modified: Mo 01.09.2025 11:46