Summability Methods and Gabor Analysis

Summability methods constitute a central part of classical Fourier analysis, arising from the observation that the ordinary partial sums of the Fourier series of a function of one or several variables may show a bad behavior, which may be very much ameliorated by the use of summability methods. More generally, the difficulties with the direct form of the inversion formula can be avoided by premultiplying the Fourier transform with suitable parametrized "kernels", and letting then the kernel parameter tend to a limit. Although it was considered a classical branch of analysis there was and is still a lot to do, as the applicants work in the last years shows. In the last 20-30 years a new section in summability theory has been opened by the application of martingale and classical Hardy spaces and their atomic decomposition and the one of the interpolation theory. It was possible in this way to investigate summations of one- and several-dimensional Fourier series and so almost everywhere convergence theorems have been obtained. The book of Weisz (2002) contains a large collection of recent results and gives a good overview of this topic. On the other hand, Gabor analysis has established itself in the last 20 years as a fertile branch of analysis, with a lot of links to real world applications. Basically, one may view it as local Fourier theory, allowing to perform analysis of functions showing a variable behavior over time (in contrast to Fourier series which are only suitable for periodic functions, or the Fourier transform which only gives a global picture of the frequency distribution within a function). The books by Feichtinger/Strohmer (1998,2002) or Gröchenig (2001) are good sources for a researcher interested in this branch of analysis. There are many natural connections between summability theory and Gabor analysis, and this project would allow to develop these connections, with a very good chance for interesting results giving new insights for both subdisciplines.