Balian-Low phenomena, exponential bases and neural network approximation

Kurzfassung/Abstract

The present cumulative thesis is based on 6 papers concerning 3 topics: Balian-Low theorems, exponential bases and neural network approximation.

Various extensions of the Balian-Low theorem are proved for Gabor spaces spanning strict subsets of the space of square-integrable functions on the real line.

Concerning exponential bases, we show that a partition of the unit interval into subintervals with rationally independent endpoints admits a so-called hierarchical Riesz spectral set.

Finally, in terms of neural network approximation, quantitative rates for the approximation of complex valued functions with Sobolev (real) regularity via complex valued neural networks are obtained. Additionally, a network architecture is provided for approximating classifier functions with Barron boundaries without the curse of dimensionality.