details to be given soon (27.12.2010) hgfei

Time-frequency analysis (known under many different names, such as Gabor Analysis, coherent states, etc.) is a possible way approach to many questions in analysis from a view-point which gives the time-representation of a signal the same relevance as the frequency representation. Even more than classical Fourier analysis the relatively simple algebraic situation that one has over FINITE ABELIAN GROUPS requires a more elaborative (but still very natural) functional analytic approach. Since there are no decent orthonormal bases (unlike wavelet theory) of Gabor type (due to the Balian-Low theorem) on has to accept to work with Banach frames, while on the other hand the question of best approximation of a given operator (in the Hilbert-Schmidt sense) naturally leads to the consideration of Riesz (projection) basis of the corresponding rank-one operators.

It turns out the TF-analysis not only poses a number of delicate questions, but also opens the way to a new Banach Gelfand Triple (i.e. a triple, consisting of a Banach spaces of test-functions, sometimes called Feichtinger's algebra, and the corresponding dual space, with the Hilbert space L2(G) in the middle).

It is the purpose of this talk to demonstrate how easy it is tu use this Banach Gelfand Triple, how it helps to give a precise mathematical meaning to terms arising in Gabor analysis, and to extend the algebraic facts valid over finite Abelian groups to the setting of general LCA groups.