Course content

The course presents an introduction to applied harmonic analysis covering the basics of time-scale analysis and time-frequency analysis. The continuous wavelet transform and the short-time Fourier transform are discussed as well as its discretization, which concerns the construction and properties of wavelet bases and Gabor frames. In this context, the relevant mathematical concepts are reproducing kernel Hilbert spaces, frames, and Riesz bases for Hilbert spaces. Applications related to areas such as signal analysis, image processing, and machine learning will be also discussed.

Learning outcome

1. Knowledge: The course presents an introduction to applied harmonic analysis covering the basics of time-scale analysis and time-frequency analysis. The continuous wavelet transform and the short-time Fourier transform are discussed as well as its discretization, which concern the construction and properties of wavelet bases and Gabor frames. In this context, the relevant mathematical concepts are reproducing kernel Hilbert spaces, frames and Riesz bases for Hilbert spaces. Applications related to areas such as signal analysis, image processing and machine learning will be also discussed.

2. Skills: The students should be able to handle problems and conduct research related to theoretical and applied problems related to wavelet theory, and, more generally, time-frequency analysis. In particular, techniques connected with signal and image processing should be studied.

3. Competence: The students should be able to participate in scientific discussions and conduct research on high international level in wavelet theory, time-frequency analysis and its applications as well as to collaborate in joint interdisciplinary research projects.

Learning methods and activities

Lectures, alternatively guided self-study.

The course is taught every second year, if there are enough students, next time Fall semester 2025. If there are few students, there will be guided self-study.

Recommended previous knowledge

Fourier analysis course, or at least Mathematics 4.

Course materials

Will be announced at the start of the course.