Gabor frames, unimodularity, and window decay
Given \(a,b>0\) and a function \(g\in L^2(R)\), consider the collection of functions \(g_{na,mb}(t):= e^{2\pi i mbt}g(t-na), \;m,n \in Z.\) Suppose that \(\{g_{na,mb}\}_{m,n\in Z}\) is a frame for \(L^2(R)\) and let \(\gamma^0\) be the minimal dual. Then \(f= \sum_{m,n} (f, \gamma^0_{na,.mb})g_{na,mb}\) for all \(f\in L^2(R)\). In order to apply this result, it is important to know to what extent \(\gamma^0\) inherits decay properties of \(g\). In the general case it is proved that exponential decay of \(g\) implies exponential decay of \(\gamma^0\). By time-frequency symmetry it follows that exponential decay of \(g\) in time and frequency implies the same property for \(\gamma^0\). In case of a rationally oversampled frame, a necessary and sufficient condition for a compactly supported function \(g\) to have a compactly supported minimal dual is given.
scientific article; zbMATH DE number 1445611
scientific article