Hans Georg Feichtinger (Vienna University):
Title: Banach Frames and Banach Gelfand Triples
Abstract:
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.