Florian Le Manach | Searcharxiv

arXiv2018

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

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 $\ell^p(\mathbb{Z})$, $1<p<2$.

math.FA↗

arXiv2017

Cyclicity in weighted $\ell^p$ spaces

We study the cyclicity in weighted $\ell^p(\mathbb{Z})$ spaces. For $p \geq 1$ and $β\geq 0$, let $\ell^p\_β(\mathbb{Z})$ be the space of sequences $u=(u\_n)\_{n\in \mathbb{Z}}$ such that $(u\_n |n|^β)\in \ell^p(\mathbb{Z}) $. We obtain both necessary conditions and sufficient conditions for $u$ to be cyclic in $\ell^p\_β(\mathbb{Z})$, in other words, for $ \{(u\_{n+k})\_{n \in \mathbb{Z}},~ k \in \mathbb{Z} \}$ to span a dense subspace of $\ell^p\_β(\mathbb{Z})$. The conditions are given in terms of the Hausdorff dimension and the capacity of the zero set of the Fourier transform of $u$.

math.FA↗