Sparse Shearlet Representation of Fourier Integral Operators

Fourier Integral Operators appear naturally in a variety of problems related to hyperbolic partial differential equations. While wavelets and other traditional time-frequency methods have been successfully employed for representing many classes of singular integral operators, these methods are not equally effective in dealing with Fourier Integral Operators. In this paper, we show that the shearlets provide a very efficient tool for the analysis of a large class of Fourier Integral Operators. The shearlets, recently introduced by the authors and their collaborators, are an affine-like system of well-localized waveform at various scales, locations and orientations. It turns out that these waveforms are particularly adapted to the action of Fourier Integral Operators. In particular, we prove that the matrix representation of a Fourier Integral Operator with respect to a Parseval frame of sherlets is sparse and well-organized. This fact confirms similar results recently obtained by Candès and Demanet, pointing out to the effectiveness of appropriately constructed directional multiscale representations in dealing with operators associated with hyperbolic problems.