We consider a continuous version of Gabor multipliers: operators consisting of a short-time Fourier transform, followed by multiplication by a distribution on phase space (called the Gabor symbol), followed by an inverse short-time Fourier transform, allowing different localizing windows for the forward and inverse transforms. For a given pair of forward and inverse windows, which linear operators can be represented as a Gabor multiplier, and what is the relationship between the (non-classical) Kohn-Nirenberg symbol of such an operator and the corresponding Gabor symbol? These questions are answered completely for a special class of "compatible" window pairs. In addition, concrete examples are given of windows that, with respect to the representation of linear operators, are more general than standard Gaussian windows. The results in the paper help to justify techniques developed for seismic imaging that use Gabor multipliers to represent nonstationary filters and wavefield extrapolators.
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