Hermite Multiplikator, Faltungen und Zeit-Frequenz-Analyse

Wider research context: The concept of convolution is naturally linked with the Fourier transform in its various forms (over locally compact Abelian groups). In this project it is planned to study variants of this concept which are both making the link between the Hermite functions and Time-Frequency analysis. The project also combines the topics of Gabor and Hermite expansions of tempered distributions. The study of Gabor multipliers or Anti-Wick operators, or Hermite-(pseudo)-multipliers utilize the corresponding Banach spaces of distributions, known as modulation spaces or Hermite-Besov-spaces, which allow to describe the mapping properties of the corresponding multipliers. We will study their mutual approximations and their action on the function spaces defined by the other family expansion type or by discrete versions.

Objectives: First we study operators which can be described in both ways, and then study cases where the description in one system is easy to understand while the behaviour in the other setting is not at all obvious. The prototypical family of examples in this direction is the Fractional Fourier transforms. A more applied aspect of the project are 'discrete versions of these operators'. We will compare different approaches taken of this kind and study their behaviour over finite domains as opposed to the Euclidean domain.

Approach: The approach will be based on a combination of methods from the theory of function spaces and computational harmonic analysis, from Gabor and Hermite Analysis. The use of modulation spaces, the Banach Gelfand Triple and specific properties of Feichtinger's algebra will play significant role. The numerical approximation of Hermite and Gabor multipliers on the different spaces will be based on compactness arguments.

Level of innovation: Despite existing literature on both sides (Hermite multipliers or pseudo-multipliers respectively time-frequency and Gabor Analysis) the connections of these two areas is still at an early stage and is expected to offer a fruitful area of research. The far goal of such investigations is the combination of efficient code combined with a theoretical backup, which then will allow to solve a problem numerically, up to a given error in a suitable function space norm. The project aims to contribute towards 'Conceptual Harmonic Analysis' as promoted by the host in the last years.

Primary researchers involved: Dr. A. Gumber, currently a PostDoc at NuHAG (Faculty of Math., University of Vienna), is the main person carrying out the research in cooperation and under the guidance of the host, H. Feichtinger. Since the NuHAG is an international key-player in time-frequency analysis several members from the team will be involved in the research project. Concrete cooperation is planned with M. Dörfler, M. Faulhuber, and M. de Gosson. External cooperation partners are P. Boggiatto, F. Filbir, F. Luef and S. Thangavelu.