Banach Frames and Banach Gelfand Triples

Frames are a well established and widely used concept within harmonic analysis and modern signal processing, and they are of relevance in the context of time-frequency analysis (which does not allow for orthogonal bases of Gaborian type, unlike wavelets), but also in wavelet theory, the study of the shearlet transform and other continuous frames. One of the points of the discussion is the emphasize that one should not restrict the attention to Hilbert spaces only, but keep in mind, that each expansion allows for a variety of ambient Banach spaces, and that in this context atomic decompositions and Banach frames have to be used. They are more than just norm-equivalences for individual Banach spaces. In order to make things not too general one may restrict the attention to so-called Banach Gelfand triples, which consist of a Banach space (of test functions) sitting inside the Hilbert space, 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 Banach Gelfand Triple isomorphisms.