In a number of earlier talks and papers I have tried to popularize the view-point that
typical unitary mappings arising in Fourier analysis, such as the (generalized) Fourier
transform, the kernel theorem establishing a connection between the space of all
Hilbert Schmidt operators on $\LtRdN$ and its $\LtRtd$-integral kerrnel, or the Weyl-calculus building operators from symbols over phase space should better be interpreted as (unitary) Banach Gelfand triple isomorphisms between very natural rigged Hilbert spaces or Banach Gelfand Triples (BGT) surrounding the central Hilbert spaces, arising in those different settings. Specifically the BGT arising from the Segal algebra SO(Rd) and it's dual SO'(Rd) are useful in time-frequency analysis.

In our talk we want to point out that a number of usual constructions can be
carried out within the family of so-called standard spaces which are
sitting in between SO(Rd) and SO'(Rd) e.g. taking the Fourier image,
defining spaces of pointwise or Fourier multipliers, or constructing Wiener
amalgam spaces base on a given standard space. For each such space one can
find a minimal and a maximal space with the same norm, which can be characterized using regularizing families, namely the closure of the test functions and the
vague relative completion of the given space. Reflexive space can be
characterized by the property that both the space and its dual are at
the same time minimal and maximal (a well known situation if one considers
Lp-spaces).
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In principle we want to reconnect earlier work in with W.Braun
with more recent observations concerning Banach Gelfand triples.