Intrinsic localization of anisotropic frames

The present article studies off-diagonal decay properties of Moore–Penrose pseudoinverses of (bi-infinite) matrices satisfying an analogous condition. Off-diagonal decay in our paper is considered with respect to specific index distance functions which incorporates those usually used for the study of localization properties for wavelet frames but also more general systems such as curvelets or shearlets. Our main result is that if a matrix satisfies an off-diagonal decay condition, then its Moore–Penrose pseudoinverse satisfies a similar condition. Applied to the study of frames this means that, if a wavelet, curvelet or shearlet frame is intrinsically localized, then its canonical dual is, too.