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

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: weighted $\ell^p$ spaces; cyclicity

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

Florian Le Manach. Cyclicity in $\ell^p$ spaces and zero sets of the Fourier transforms. Journal of Mathematical Analysis and Applications, 2018, Journal of Mathematical Analysis and Applications, 462 (1), pp.967-981. ⟨10.1016/j.jmaa.2017.12.057⟩. ⟨hal-01570349v2⟩