It is clear that the theory of tempered distributions is in some sense limited, because it does not allow to define the Fourier transform for objects which have exponential growth. On the other hand the Schwartz-Bruhat theory, which is the analogue of L.Schwartz theory of tempered distributions for general LCA (locally compact Abelian) groups is already quite cumbersome.

Since the borderline between Banach spaces of test functions which are smaller and smaller and the trivial one (the useless space, consisting only of the zero-function) is quite thin it is interesting to take a look at the relevant condition, which is formulated in the classical Acta Mathematica paper by Yngve Domar entitled Harmonic analysis based on certain commutative Banach algebras. In this paper the so-called Beurling-Domar condition is formulated, which also appears in the book of Hans Reiter.

We will discuss various aspects of this approach and to which extent it allows to build Banach spaces of test functions (and corresponding distributions), and even finally Frechet spaces of this type, over general LCA groups. We hope to indicate that this view-point allows certain unifications and technical simplifications compared to the technical details needed either in the Schwartz-Bruhat or in the now classical theory of ultra-distributions over the Euclidean space.