Taking the current viewpoint classical Fourier Analysis
started from the study the orthogonal expansion of
periodic functions in the Hilbert space L^2(T),
using the orthonormal system of pure frequencies
(described by complex exponential functions).

Moving on to the real line it was natural to view
the Fourier transform, given by the usual integral
formula, as a linear mapping defined on the Lebesgue
space L^1(R), mapping into the space C_0(R) of
continuous functions, vanishing at infinity (by the
Riemann-Lebesgue Lemma). The convolution theorem then
goes on and describes this mapping as an injective
Banach algebra homomorphism, converting convolution
into pointwise multiplication. This is also the basis
for the study of summability methods which are used
in order to verify the validity of Fourier inversion
formulas.

As it turned out, L^p-spaces are not so well suited
for the study of the Fourier transform (only L^2 is
invariant) and even less for the modern theory of
Time-Frequency Analysis (TFA) or Gabor Analysis (a
branch of TFA dealing with discrete representations).
Instead, modulation spaces are better suited. The
prototypes in this family are the Segal algebra
S_0, also called Feichtinger's algebra, and its dual,
meanwhile known as Banach space of mild distributions.
Together with the Hilbert space L^2 they form THE
BGT: Banach Gelfand Triple (SO,L2,SO*), which behaves
in many ways like the triple (Q,R,C) of Q (rational),
R (real) and (C: complex) numbers. Although these
spaces can be defined and are useful in the context
of general locally compact Abelian (LCA) groups we
will discuss a (long) list of features making them
extremely useful, for the study of both theoretical
and applied (say in signal processing) problems.

THE BGT allows for an elegant derivation of the basic
principles of Fourier Analysis, but it is also a
central tool for TFA and Gabor Analysis. It also
provides a solid foundation of a mind-set called
Conceptual Harmonic Analysis (CHA). Although it is
persued by the author now for several years, it still
lacks popularity, competing with traditional approaches.

Overall CHA suggests to promote the integration of
concepts from basic functional and harmonic analysis
with numerical computations via efficient algorithms.
It also supports the modelling real world problems
(such as recovery of signals from sampling), thus
avoiding purely heuristic ``hand-waving'' arguments.