Balian-Low Type Theorems for Shift-Invariant Spaces

Abstract

Shift-invariant spaces and Gabor systems are frequently used in approximation theory and signal processing. In these settings, it is advantageous for the generators of such spaces to be localized and for the spaces to be representative of a large class of functions. For Gabor systems, the celebrated Balian-Low Theorem shows that if the integer translations and modulations of a function in L^2(R) form an orthonormal basis for L^2(R), then either the function or its fourier transform must be poorly localized. The present work shows that similar results hold in certain shift-invariant spaces. In particular, if the integer translates of a well-localized function in L^2(R^d) form a frame for the shift-invariant space, V, generated by the function then V cannot be invariant under any non-integer shift. Similar results are proven under a variety of different basis properties, for finitely many generating functions, and with the extra-invariance property replaced with redundancy in the translates of the generator. Examples are given showing the sharpness of these results.