Fall 2025



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



Lectures on Nov. 24th: 10:15-12:00 and 13:15-15:00 in Simastuen.

Papers for the presentation for the exam

  • L.D. Abreu and M. Doerfler. An inverse problem for localization operators. Inverse Problems, 28, 115001 (2012). pdf
  • F. Luef and E. Skrettingland. Convolutions for localization operators. J. Math. Pure Appl. 118, 288–316 (2018). pdf
  • M. Appleby, I. Bengtsson, S. Flammia, and D. Goyeneche. Tight frames, Hadamard matrices and Zauner's conjecture. J. Phys. A 52, no. 29, 295301, 26 pp (2019). pdf
  • I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal Gabor expansions. J. Math. Phys. 27, no. 5, 1271–1283 (1986). pdf
  • M. Basarab, Y. Meyer, and J. Ortega-Cerda. Stable sampling and Fourier multipliers. Publ. Mat. 58, no. 2, 341–351 (2014). pdf



This course gives an introduction into some of the basic concepts of applied harmonic analysis including convolutional neural networks, phase retrieval, reproducing kernel Hilbert spaces, wavelets, Gabor frames and quantum harmonic analysis, which provides extensions of the notions discussed for functions in the course to operators. The latter is of relevance for detecting local structures in the spectral behavior of time-series.

The aim is to prepare you for working with signals, images and data sets ranging from standard to state of the art tools relevant for machine learning and other modern aspects of applied harmonic analysis.

"The theory of wavelets lies on the intersection of (1) mathematics (2) scientific computation (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)

References

  • O. Christensen. An introduction to Frames and Riesz bases. Springer link
  • K. Groechenig. Foundations of time-frequency analysis. Birkhaeuser link
  • I. Daubechies. Ten lectures on wavelets. Wiley see blackboard



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

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



  • 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)
  • There will be lecture notes for the material covered in the course.
2025-11-25, Franz Luef