Representation of Linear Operators by Gabor Multipliers

Abstract

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. This chapter focuses on the following broad questions. Firstly, 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 Kohn–Nirenberg symbol of such an operator and the corresponding Gabor symbol? We answer this question completely. Secondly, for a linear operator of a given type, can windows be specially chosen, or “tuned”, to suit the operator so that the Gabor symbol reflects the operator’s type? In studying this latter question for product-convolution operators, we derive a new class of “extreme value” windows that, with respect to the representation of linear operators, are more general than standard Gaussian windows while sharing many of Gaussian windows’ desirable properties. The results in this chapter help to justify techniques developed for seismic imaging that use Gabor multipliers to represent nonstationary filters and wavefield extrapolators.