Universität Wien

250087 VO Frame Theory (2021W)

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

  • Tuesday 05.10. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 07.10. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 12.10. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 14.10. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 19.10. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 21.10. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Thursday 28.10. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Thursday 04.11. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 09.11. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 11.11. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 16.11. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 18.11. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 23.11. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 25.11. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 30.11. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 02.12. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 07.12. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 09.12. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 14.12. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 16.12. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 11.01. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 13.01. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 18.01. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 20.01. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.Stock
  • Tuesday 25.01. 11:30 - 12:15 Seminarraum 14 Oskar-Morgenstern-Platz 1 2.Stock
  • Thursday 27.01. 13:15 - 14:45 Seminarraum 16 Oskar-Morgenstern-Platz 1 3.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. Particular The implementation of frame-related algorithms will be considered and 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

Written exam

(In exceptional cases an oral exam is possible.)

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

MANV; MAMV

Last modified: We 01.02.2023 00:26