  A new approach to problems of the Uncertainty Principle in Harmonic Analysis,
based on the use of Toeplitz operators, has brought progress to some of the
classical problems in the area. The goal of this paper is to develop and
systematize the function theoretic component of the Toeplitz approach by
introducing a partial order on the set of inner functions induced by the action
of Toeplitz operators. We study connections of the new order with some of the
classical problems and known results. We discuss remaining problems and
possible directions for further research.
