Abstract:
For g in L^2(R) we define the ab-Gabor system, (g,a,b), to be the sequence g_mn={ g(x-na)e^{2 \pi i m b x} } for all m,n in N. A system is said to be oversampled if ab<1. Two of the main tools used in analyzing Gabor systems are the Zak transform Z:L^2(R) -> L^2([0,1]x[0,1])
Z(f)(t,v)=\sum_k f(t-k) e^{2 \pi i k v}
and the pointwise inner product (also known as the bracket product)
<f,g>_a(x)= \sum_k f(x-ka)\overline{g(x-ka)}.
The Zak transform is particularly effective in the case when ab=1 , i.e. where oversampling does not occur. Most of the results can be mimicked for systems where ab<1 but require this product ab to be rational. We combine these two tools to produce generalizations of the Zak transform geared toward analyzing oversampled systems.