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

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, $\mathcal{Z}(\hat{u})$, of its Fourier transform, $\hat{u}$, has Lebesgue measure zero and $\log |\hat{u}|\notin L^1(\mathbb{T})$, 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 $\mathcal{Z}(\hat{u})$ and the divergence of the integral $\int \_\mathbb{T}\log |\hat{u}|$. Moreover we give both necessary conditions and sufficient conditions for $u$ to be cyclic in ${\ell }^p(\mathbb{Z})$, $1<p<2$.

Australian Journal of Mathematical Analysis and Applications, Austral Internet Publishing, A Para{î}tre

References in corpus (2)

• Wiener's 'closure of translates' problem and Piatetski-Shapiro's uniqueness phenomenon
• Cyclicity in the harmonic Dirichlet space