Resolution of Sobolev wavefront set and sparse representation of singular integral operator using shearlets
The main ingredients of time-frequency analysis, such as wavelets and Gabor frames have been successfully used for the representation of most of the classes of pseudo-differential operators, singular integral operators (SIOs). The location and the geometry of the set of singularities of a distribution can be obtained by using continuous transform. The continuous transform of a distribution is related to the microlocal analysis. Microlocal analysis can be employed to study how singularities propagate under certain classes of operators, i.e., Fourier integral operators, pseudo-differential operators as well as many integral operators arising in integral geometry. Wavefront set of a distribution is an essential concept in the microlocal analysis. The microlocal analysis is perhaps useful in inverse problems, where the goal is to recover the wavefront set of a function or distribution from the solution of operator equation. In this talk, our aim is to characterize sobolev wavefront set using shearlets, and its connection with Holder regularity. Later we show that the shearlets provide very efficient representations for a large class of SIO. Shearlets are particularly useful in representing anisotropic functions due to the properties of an affine-like system of well-localized waveforms at various scales, locations, and orientations. This is a joint work with Dr. Niraj K. Shukla.
2019-06-21 at 2:30 pm (subject to variability)