Cyclicity in $\ell^p$ spaces and zero sets of the Fourier transforms

TL;DR

This paper investigates the conditions under which vectors in lp(Z) are cyclic, revealing that the characterization known for l2(Z) does not extend to 1<p<2, and providing new necessary and sufficient conditions.

Contribution

It demonstrates the lack of a complete characterization of cyclicity in lp(Z) for 1<p<2 and offers new criteria for cyclicity in this range.

Findings

No full characterization of cyclicity for 1<p<2 in terms of zero sets and integrability.

Necessary conditions for cyclicity in lp(Z) for 1<p<2.

Sufficient conditions for cyclicity in lp(Z) for 1<p<2.

Abstract

We study the cyclicity of vectors $u$ in $\ell^p(\mathbb{Z})$. It is known that a vector $u$ is cyclic in $\ell^2(\mathbb{Z})$ if and only if the zero set, $\mathcal{Z}(\widehat{u})$, of its Fourier transform, $\widehat{u}$, has Lebesgue measure zero and $\log |\widehat{u}| \not \in L^1(\mathbb{T})$, where $\mathbb{T}$ is the unit circle. Here we show that, unlike $\ell^2(\mathbb{Z})$, there is no characterization of the cyclicity of $u$ in $\ell^p(\mathbb{Z})$, $1<p<2$, in terms of $\mathcal{Z}(\widehat{u})$ and the divergence of the integral $\int\_\mathbb{T} \log |\widehat{u}| $. Moreover we give both necessary conditions and sufficient conditions for $u$ to be cyclic in $\ell^p(\mathbb{Z})$, $1<p<2$.