Uncertainty principles for the affine group

Abstract

The Lie algebra of the affine group is generated by two operators $A$ and $B$ satisfying the commutator rule $ [A,B] = B$. A version of the uncertainty principle is designed such that - in the time domain - the extremal functions are real valued. The uncertainty inequality naturally contains a parameter. In the application the wavelet transform based on the extremal functions gives a model for the first stage of the hearing perception in the inner ear (the cochlea). The parameter in the uncertainty inequality is associated to the position along the cochlea.