The key question of Gabor Analysis can essentially be formulated in to more or less equivalent version, both making use of the sampled short-time Fourier transform (STFT). We discuss the case $R^d$.

In order to define the STFT of a tempered distribution (say) with respect to some window in the Schwartz space of rapidely decreasing functions one moves the window over the signal (by shifting it within the underlying group $R^d$) and then take, at any position a Fourier transform, which describes the local behaviour of the function or distribution. For one-dimensional signals this is comparable to a musical score, proding information about the harmonic engergy distribution within a signal, as it varies over time.

The two viewpoints are:
A) recover the STFT from a sampled version, preferably over a lattice in phase space, so in $R^{2d}$; or
B) Given a set of Gabor atoms, i.e. TF-shifted version of some Gabor atom, how can one expand a given function or distribution into a unconditionally convergent series, using these building blocks,
and how can one determine a suitable set of coefficients.

In both case we talk about Banach frames of Gaborian type, and one possible answer is to use the dual frame.

Even if one is interested only in $L^2$ results there are many good reasons for invocing the Banach Gelfand triple $(S_0,L_2,S_0')$,
consisting of the Segal algebra $S_0(R^d)$ (also called Feichtinger's algebra), the Hilbert space $L_2$ and the dual space.

In the talk we will explain why and how this triple can be used and arises naturally in TF-analysis, but it is also very useful in the context of classical Fourier Analysis.


Host: Dr. S K Upadhyay
Department Mathematical Science
Indian Institute of Technology
(Banaras Hindu University)
Varanasi, 221 005