SHEARLET TRANSFORMS AND DIRECTIONAL REGULARITIES

摘要

In an effort to characterize uniform and pointwise Holder regularities, we obtain necessary decay rates and sufficient decay rates of continuous and discrete shearlet transform across scales. They are the same rates as those of the Hart Smith and continuous curvelet transforms. We then consider the situation where the regularity on a line in a non-parallel direction is significantly lower than the directional regularity along the line in a neighborhood of the line. Similar to that of the Hart Smith and continuous curvelet transforms, a set of necessary conditions for this direction of singularity is that the continuous shearlet transform decays half an order faster in directions "away" from the direction of the line and that the decay rate in directions "near" the line depends also on the horizontal distance from the line to the parallel line containing the center of the shearlet function. These similarities among the three transforms call for a general theory of transforms "with parabolic scaling."