Square Integrable Group Representations and the Uncertainty Principle

Let U be a square integrable representation of a Lie group G of transformations in a Hilbert space H, and let Ψ ∈ H be an admissible state. We call the product of variances in the state ψ, associated to two non-commuting infinitesimal operators T₁ and T₂, uncertainty measure. We investigate, in this note, how uncertainty measures of admissible states ψ are changing when ψ is transformed according to U(g)ψ for g∈G. We derive these transform laws for certain types of the associated Lie-algebra and apply them to relevant transform groups (affine group, similitude group, shearlets), investigated recently in signal processing. By doing so, we are able to show that the associated uncertainty measures can be made arbitrarily small in many cases. This implies that the corresponding "uncertainty minimizers", which are traditionally defined via equality in the famous uncertainty inequality and which have been investigated recently for above mentioned groups, actually do not minimize the uncertainty measure. In cases, where the uncertainty measure of U(g)ψ cannot be made arbitrarily small, when g varies over G, the transform laws derived in this paper indicate how to pick group elements g′ such that the uncertainty measure of U(g′)ψ attains a local minimum. This method potentially enables discretizations of square integrable group representations with "low uncertainty", or equivalently "high locality" of the features of interest. This is of great importance for connecting uncertainty principles with frame constructions, and for maximum accuracy of feature measurements in signals and images.