Intrinsic Localization of Anisotropic Frames II: alpha-Molecules

摘要

This article is a continuation of the recent paper (Appl Comput Harmon Anal 35:264-283, 2013) by the first author, where off-diagonal-decay properties (often referred to as 'localization' in the literature) of Moore-Penrose pseudoinverses of (bi-infinite) matrices are established, whenever the latter possess similar off-diagonal-decay properties. This problem is especially interesting if the matrix arises as a discretization of an operator with respect to a frame or basis. Previous work on this problem has been restricted to wavelet- or Gabor frames. In Appl Comput Harmon Anal 35:264-283, 2013, we extended these results to frames of parabolic molecules, including curvelets or shearlets as special cases. The present paper extends and unifies these results by establishing analogous properties for frames of -molecules as introduced in recent work (Proc SPIE, 2013). Since wavelets, curvelets, shearlets, ridgelets and hybrid shearlets all constitute instances of -molecules, our results establish localization properties for all these systems simultaneously.