Link between cyclicity of a vector in $\ell^p$ spaces and zero set of its Fourier transform

Florian Le Manach (1)

1 IMB - Institut de Mathématiques de Bordeaux

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

Mots clés: cyclicity; weighted $\ell^p$ spaces

Domaines: Analyse fonctionnelle [math.FA]; Analyse classique [math.CA]

Florian Le Manach. Link between cyclicity of a vector in $\ell^p$ spaces and zero set of its Fourier transform. 2017. ⟨hal-01570349v1⟩