The aim of this seminar is to discuss both the theoretical and numerical aspect of wavelets.

Wavelets are basis functions with some additional properties and are often used in multi-scale analysis. Similarly to Fourier techniques signals can be decomposed in frequencies, which is useful in signal processing to remove noise.
Additional applications of wavelets can be found, e.g., in image compression.

Mathematically wavelets are orthonormal bases constructed by dilation and translation of a so called mother wavelet. An advantage compared to classical sinus/cosinus-bases is that wavelets are not only lokal in the frequency domain but also local in the time domain.

The aim of this seminar is to discuss both the theoretical and numerical aspect of wavelets. We will study wavelet Galerkin methods, which lead to quasi-sparse matrices, as well as the theoretical connection between smoothness of functions and approximability by wavelets.

Wavelets are an ongoing research topic, for which Yves Meyer was awarded the Abel price in 2017.