Host: Prof. Pankaj Jain

The classical approach to Fourier analysis emphasises the importance of Lebesgue spaces, because they are important in order to define the Fourier transform as an integral transform, the same for the inverse Fourier transform, defined on $L^1$. Also convolution (defined in a pointwise sense, a.e.) appears to call for Lebesgue integration.
The Hilbert space $L^2$ is the correct tool in order to describe the energy preserving property of the Fourier transform.

However, there are problems, even in the classical setting, and if one goes to Gabor analysis the $L^p$ spaces completely loose their relevance (except for $L^2$). Thus other function spaces are needed.
We plan to describe the setting of Gabor analysis, the claim of D.Gabor (from 1946) that every function has a unique representation as an infinite double sum, with building blocks which are time-frequency shifted copies of a Gaussian function.

Using the function space $S_0$ and its dual $S_0'$ one can form a Banach Gelfand triple with $L^2$ as the Hilbert space in between the space of test functions $S_0$ and the corresponding (mild) distributions, forming $S_0'$. Even the classical Fourier transform
has a nice interpretation in this setting.