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
_ψ
u(x, h) = 〈u|T
_x
D
_h
ψ〉, where the wavelet ψ is dilated by elements h ε H of a certain dilation group H ≤ GL (ℝ
^d
). We provide readily verifiable, geometric conditions on the dilation group H which guarantee that (χ,ξ) is a regular directed point of u iff the wavelet transform W
_ψ
u has rapid decay on a certain set depending on (x, ξ). Roughly speaking, smoothness of u near x in direction ξ is equivalent to rapid decay of wavelet coefficients W
_ψ
u (y, h) for y near x if D
_h
ψ is a small scale wavelet oriented in a direction near ξ. 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
.