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 shiftinvariant spaces on the real line that are also 1 n Z-invariant with an integrable orthonormal (or a Riesz) generator φ, but φ satisfies ∫ R |φ(x)||x|dx = ∞ for any ǫ > 0 and its Fourier transform φ̂ cannot decay as fast as (1 + |ξ|) for any r > 1 2 . Examples are constructed to demonstrate that the above decay properties for the orthormal generator in the time domain and in the frequency domain are optimal.