(Abstract Harmonic Analysis)} is able to understand the
similarity between different notions of the Fourier transforms
from an axiomatic viewpoint, mostly by {\it analogy}.

Given a {LCA group $\cG$} it is well known that there
is a translation invariant linear functional on $\CcG$,
called the {Haar measurer}. Consequently there
is an associated Banach space $\LiGN$, which in fact is
a Banach algebra with respect to convolution, turning $\LiGN$
into a commutative Banach algebra.

There is the dual group $\cGd$, of all { characters}
of $\cG$, and once more $\LiGN$ appears as a natural domain
for the Fourier transform, which maps $\LiGN$ into $\COspN$
\newline (by the Riemann-Lebesgue Lemma).
In contrast, (Numerical \newline Harmonic Analysis) NHA provides
efficient code to realize the FT numerically in the finite, discrete setting (FFT).

BUT WE WILL VIEW THINGS FROM THE POINT OF VIEW OF the Banach Gelfand Triple, based on the Segal algebar $S_0$, its dual and the
Hilbert spaces $L2Rd$ in the middle. They form a rigged Hilbert space or THE Banach Gelfand Triple.