The talk describes a surprisingly rich family of function spaces which can be defined on general LCA groups
(locally compact Abelian groups, such as G = R^d).
The starting point is the Banach Gelfand triple (S0,L2,S0'), consisting of the Segal algebra S0(G) as a space of
test functions and the dual space as the minimal respectively maximal space in this family.
One of the most attractive (and surprising) facts about this setting, which requires only the use of
Banach spaces and their dual spaces, is the existence of a kernel theorem,
which extends the classical association of L2-kernels with the family of Hilbert-Schmidt operators.

As time permits a number of questions arising from classical analysis and time-frequency
analysis respectively Gabor analysis will bementioned.