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

TL;DR

This paper explores the limitations on the time-frequency localization of generators in principal shift-invariant spaces with additional invariance, establishing non-integrability and decay constraints that are proven to be optimal.

Contribution

It proves new uncertainty principles and Balian-Low type theorems for shift-invariant spaces with extra invariance, revealing fundamental localization restrictions.

Findings

Translation-invariant spaces have non-integrable orthonormal generators.

Existence of shift-invariant spaces with integrable generators that have slow decay.

Constructed examples demonstrate optimality of decay and localization bounds.

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 $n\ge2$, there exist principal shift-invariant spaces on the real line that are also $\nZ$-invariant with an integrable orthonormal (or a Riesz) generator $\phi$, but $\phi$ satisfies $\int_{\mathbb R} |\phi(x)|^2 |x|^{1+\epsilon} dx=\infty$ for any $\epsilon>0$ and its Fourier transform $\hat\phi$ 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.