Banach Gelfand Triples (and modulation spaces from the coorbit view-point)
Hans G. Feichtinger
given at Dept. Math., Univ. of Otago, Dunedin (26.08.10 14:00)
Gelfand Triples (sometimes called rigged Hilbert spaces) are a very useful general construction that allows to talk of good functions (think of Schwartz test functions) and most general objects (such as tempered distributions) surrounding the central Hilbert spaces. Pairs of Sobolev spaces show a similar setting and are useful for the description of elleptic PDEs.
The theme of the talk is the presentation of a particular Banach Gelfand Triple, i.e. a Gelfand triple consisting of Banach spaces (actually isomorphic to the canoncial sequence spaces (l1,l2,linfty)) which arose in the context of time-frequency analysis resp. Gabor analysis. It is described by the integrability of the short-time Fourier transform of its elements (with respect to a Gaussian window). It is however not only useful in that original context, but has a wide range of applications, e.g. to give an optimal description of the Fourier transform, or allowing a kernel theorem (the analogue of a matrix representation of linear mappings between finite dimensional vector spaces), similar to the Schwartz kernel theorem.
This triple is just the prototype of a family of more general function spaces, called modulation spaces. They are quite similar to Besov or Triebel-Lizorkin and potential spaces, which are characterized among others by the continuous wavelet transform. Coorbit theory allows to view these spaces from a unifying point of view.
Material related to the topic of the talk is found at the NuHAG web-page www.nuhag.eu