Resolution of the Wave Front Set using general Wavelet Transforms

On the real line, it is well-known that (the decay of) the one-dimensional continuous wavelet transform can be used to characterize the regularity of a function or distribution, e.g. in the sense of Holder regularity, but also in the sense of characterizing the wave front set. In higher dimensions - especially in dimension two - this ability to resolve the wave front set has become a kind of benchmark property for anisotropic wavelet systems like curvelets and shearlets. Summarizing a recent paper of the authors, this note describes a novel approach which allows to prove that a given wavelet transform is able to resolve the wave front set of arbitrary tempered distributions. More precisely, we consider wavelet transforms of the form W(psi)u(x, h) = < u vertical bar TxDh psi >, where the psi wavelet is dilated by elements h is an element of H of a certain dilation group H <= GL (R-d). We provide readily verifiable, geometric conditions on the dilation group H which guarantee that (x, xi) is a regular directed point of u iff the wavelet transform W(psi)u has rapid decay on a certain set depending on (x, xi). Roughly speaking, smoothness of u near x in direction xi is equivalent to rapid decay of wavelet coefficients W(psi)u (y, h) for y near x if Dh(psi) is a small scale wavelet oriented in a direction near xi. Special cases of our results include that of the shearlet group in dimension two (even with scaling types other than parabolic scaling) and also in higher dimensions, a result which was (to our knowledge) not known before. We also briefly describe a generalization where the group wavelet transform is replaced by a discrete wavelet transform arising from a discrete covering/ partition of unity of (a subset of) the frequency space R-d.