The affine uncertainty principle in one and two dimensions (Q1904185)

This paper provides constructions of functions that minimize a certain uncertainty relation for one- and two-dimensional continuous wavelet transforms. The starting point is to interpret the continuous wavelet transform as a square integrable group representation, in the one-dimensional case of the affine group and in the two-dimensional case of the generalization of the affine group studied by J. P. Antoine et al. The uncertainty relation in question is an inequality for the commutator of the infinitesimal operators, and after identifying this inequality, the job is to find which functions yield equality. The two-dimensional result is perhaps the most interesting: It is shown that the Mexican hat wavelet, much used in image analysis, is a minimizing function of the uncertainty principle.