The wavelet transform, time-frequency localization and signal analysis
IEEE Transactions on Information Theory
Gabor frame sets for subspaces
Advances in Computational Mathematics
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The well-known density theorem for one-dimensional Gabor systems of the form {e^2^@p^i^m^b^xg(x-na)}"m","n"@?"Z, where g@?L^2(R), states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in L^2(R), or which forms a frame for L^2(R), is that the density condition ab@?1 is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set S@?R which is aZ-shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L^2(S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set E@?R with the property that the Gabor system associated with the same parameters a,b and the window g=@g"E, forms a tight frame for L^2(S).