Georgia Institute of Technology College of Sciences

Uncertainty principles are results which restrict the localization of a function and its Fourier transform. One class of uncertainty principles studies generators of structured systems of functions, such as wavelets or Gabor systems, under the assumption that these systems form a basis or some generalization of a basis. An example is the Balian-Low Theorem for Gabor systems. In this talk, I will discuss sharp, Balian-Low type, uncertainty principles for finitely generated shift-invariant subspaces of $L^2(\R^d)$. In particular, we give conditions on the localization of the generators which prevent these spaces from being invariant under any non-integer shifts.