Uncertainty principles and Balian–Low type theorems in principal shift-invariant spaces

In this paper, we consider the time-frequency localization of the generator of a principal shift-invariant space on the real line which has additional shift-invariance. We prove that if a principal shift-invariant space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any n⩾2, there exist principal shift-invariant spaces on the real line that are also -invariant with an integrable orthonormal (or a Riesz) generator ϕ, but ϕ satisfies for any ϵ>0 and its Fourier transform cannot decay as fast as (1+|ξ|)−r for any . Examples are constructed to demonstrate that the above decay properties for the orthonormal generator in the time domain and in the frequency domain are optimal.