Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces

Abstract

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