The purpose of this talk is to popularize the concepts of Banach Gelfand Triples and Fourier Standard Spaces. For any LCA (locally compact Abelian) group G the Banach Gelfand triple (SO,L2,SO')(G) can be introduced, consisting of the Segal algebra SO(G), the Hilbert space L2(G) and the dual space SO'(G), consisting of so-called ``mild'' distributions. These space form a chain of natural inclusions via SO in L2 in SO' (density in norm or in the w*-sense), and Wilson basis allow to identify the triple with the prototypical Banach Gelfand Triple (l1,l2,l^infty). Obviously, for G = T (the torus group) this is just (A(T), L2(T), PM(T)), using Wiener's algebra of absolutely convergent Fourier series, whose dual is the space of all pseudo-measures, resp.\ periodic distributions with bounded Fourier coefficients.

Fourier Standard Spaces are Banach spaces (B,||.||_B) which are sandwiched between SO(G) and SO'(G), and which are (roughly speaking) in addition isometrically invariant with respect to time-frequency shifts. In fact, SO is the smallest such space, and any space of tempered distributions with this property sits inside of SO'(G).
These space form a rich family, including Lp-spaces, Wiener amalgam spaces, modulation spaces, and many others.

In this talk we will focus on spaces of Convolvers (i.e. kernels of convolution operators) between two such Fourier Standard Spaces or Fourier Multipliers (pointwise multpliers on the Fourier transform side). In doing so we can provide a new approach to the characterization of convolvers for Lp-spaces as members of the dual of the so-called Herz algebra A_p(G). This characterization extends to a class of reflexive Banach function spaces.