Although most often implicitly it is fair to claim that Banach Frames and Banach Gelfand Triples Frames (resp. rigged Hilbert spaces) are a well established and widely used concept within harmonic analysis and modern signal processing. They appear in the context of time-frequency analysis (which does not allow for orthogonal bases of Gaborian type, unlike wavelets), but also in wavelet theory, or in the study of the shearlet transform and other continuous frames. The viewpoint proposed in the talk is to suggest to look out for Banach spaces surrounding a central Hilbert space and formulate theorems (e.g. concerning atomic decompositions, frame expansions) using such families of Banach spaces. A triple is the minimal example of this viewpoint, with the sequence spaces (l1,l2,linf) being the prototypical example.

Concrete demonstration of this concept are provided, using a Banach space (of test functions) sitting inside the Hilbert space L2(G), which in turn is inside the dual of the Banach space (a space of distributions, typically). There is an abundance of such BGTs, even in the classical literature, and many mappings, such as the Fourier transform or the Kohn-Nirenberg symbol attached to Hilbert Schmidt operators, can be interpreted as (unitary) Banach Gelfand Triple isomorphisms.