Packing, Covering and Time-Frequency Analysis

The principal investigator of the project "Packing, Covering and Time-frequency Analysis" is Dr. Markus Faulhuber. The anticipated lifetime of the project is 3 years.

Wider research context:
The project deals with timely problems in time-frequency analysis, more particularly Gabor analysis. We seek to make progress on questions regarding optimal sampling strategies for Gabor frames, which leads to questions on energy minimization and polarization and, in the final consequence, sphere packings and coverings. Moreover, we will investigate the frame property of Hermite functions and aim to extend the known frame set region.

Research questions:
The research emerges mainly from two basic problems in Gabor analysis:
- finding optimal sampling patterns and
- when is a Gabor system a frame?
The first problem will be pursued from the perspective of optimal sphere packings and coverings and how these problems relate to optimal frame bounds for Gaussian Gabor systems. A particular question is whether the hexagonal lattice also outperforms certain periodic structures which are not lattices. We also seek to identify minimizers of lattice theta functions for important lattices and want to know under which conditions we can apply universal optimality results to Gaussian Gabor systems.
The second question deals with the frame set conjecture for Hermite functions. We seek to enlarge the frame set of Hermite functions significantly and, thus, provide new Gabor frames with Hermite window functions.

Methods:
The central tool in time-frequency analysis is the short-time Fourier transform (STFT). Sampling the STFT and reconstructing the function from the samples leads to questions on the spectral bounds of the Gabor frame operator. Tools to approach the problems include symplectic matrices and metaplectic operators as well as the Zak transform. Moreover, we employ certain sufficiency criteria to ensure that a Gabor system is a frame, such as the Janssen test. In order to rigorously link Gabor frames with sphere packings and coverings, we study questions on energy optimization.

Innovation:
The innovative aspects of the project is to build the bridge between Gabor frames and sphere packings and coverings via energy optimization problems and, thus, make the connection rigorous. Also, for the first time, the study of Gabor frame bounds for periodic configurations which are not lattices will be carried out. In order to make progress on the question of optimal frame bounds, we need to advance the state-of-the-art in the area of polarization and find minimizers of lattice theta functions.
To further enlarge the frame set of Hermite functions, combinations of different sufficiency criteria for Gabor systems to constitute a frame will be used.

Primary researches involved:
The primary researchers involved are Dr. Markus Faulhuber and Prof. Frank Vallentin (U Cologne) who will be a collaboration partner for certain tasks related to sphere coverings.