# MA8104 Wavelets

**Messages:**

- Schedule:

Tuesday: 12:15-14:00, **room F3**, gamle fysikk (2nd floor) except in **week 39** and **week 44**, when it is in MA23, green building.

Thursday: 10:15-12:00, **room B3**, Oppredning/gruvedrift (3rd floor)

- Lecture notes: Lecture notes, version 13.9.2021.

# Fall 2021

Franz Luef, room 940 SB2, Franz [dot] Luef [at] ntnu [dot] no

This PhD course is given every second year, and usually 4th and 5th year (Master) students take it along with PhD students.

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

- Brief review of basic facts in Fourier analysis. Basics of time-scale and time-frequency analysis, wavelet transform and short-time Fourier transform.
- Uncertainty principles for the Fourier transform the short-time Fourier transform and the wavelet transform,
- Reproducing kernel Hilbert spaces, 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
- Frames and Riesz bases in Hilbert spaces, Gabor systems, wavelet bases.
- Duality theory of Gabor frames, Feichtinger's algebra
- Basics of quantum harmonic analysis and its relation to localization operators.
- Convolutional neural networks and quantum harmonic analysis.

"The theory of wavelets lies on the boundaries between (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)