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 $n\ge2$, there exist principal shift-invariant spaces on the real line that are also $\nZ$-invariant with an integrable orthonormal (or a Riesz) generator $ϕ$, but $ϕ$ satisfies $\int_{\mathbb R} |ϕ(x)|^2 |x|^{1+ε} dx=\infty$ for any $ε>0$ and its Fourier transform $\hatϕ$ cannot decay as fast as $(1+|ξ|)^{-r}$ 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.

No reviews yet.

No comments yet.