# MA8104 Wavelets and related areas

* Messages:* \\.

- Here is a revised version of the Lecture Notes, with more material on the Walnut representation and its relation to the Janssen representation 08.12.2023.
- Here is the material on wavelet frames and MRA pdf that I based the presentation in the lectures one.

# Fall 2023

Lecturer: Franz Luef, room 940 SB2, franz [dot] luef [at] ntnu [dot] no

**Papers for the presentation for the exam**

- L.D. Abreu and M. Doerfler. An inverse problem for localization operators. 2012 Inverse Problems, 28 115001. pdf
- F. Luef and E. Skrettingland. A Wiener Tauberian theorem for operators and functions. Journal of Functional Analysis, Volume 280, Issue 6, 2021, 108883. pdf
- I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271–1283, 1986. pdf
- R. M. Balan, B. G. Bodmann, P. G. Casazza and D. Edidin. Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl., Vol.15 No.4, (2009) p.488-501. pdf
- I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, Vol.34 No.4, (1988) p.605-612. pdf

Here is a brief description of the content of the course:

- Frames and Riesz bases in finite-dimensional Hilbert spaces
- Brief review of basic facts in Fourier analysis.
- Reproducing kernel Hilbert spaces, Bargmann-Fock spaces, Bergman spaces
- Basics of time-scale and time-frequency analysis, wavelet transform and short-time Fourier transform.
- wavelet spaces, reproducing properties of the short-time Fourier transform and the wavelet transform, links to Bargmann-Fock space and Bergman spaces.
- Localization operators and Toeplitz operators
- Construction of wavelets and Gabor frames and their basic properties
- Duality theory of Gabor frames, Balian-Low theorem, Feichtinger's algebra
- Phase retrieval
- Convolutional neural networks
- Basics of quantum harmonic analysis.

"The theory of wavelets lies on the intersection pf (1) mathematics (2) scientific calculation (3) signal processing (4) image processing. The aim of the theory is to give a coherent set of concepts, methods and algorithms met in each of these disciplines"

Yves Meyer (Abel prize laureate, 2017)

- The level is suitable for good students in the third year of study.
- It can be taken as a regular course (MA8104) or a 'fordypningsemne' (TMA4505).
- Prerequisites: TMA4170 Fourier Analysis (recommended), Matematikk 4 (obligatory)