Cyclicity in $\ell^p$ spaces and zero sets of the Fourier transforms
Language
en
Article de revue
This item was published in
Journal of Mathematical Analysis and Applications. 2018-06-01, vol. 462, n° 1, p. 967-981
Elsevier
English 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, $\widehat{u}$, ...Read more >
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 $\ell^p(\mathbb{Z})$, $1<p<2$.Read less <
English Keywords
weighted $\ell^p$ spaces
cyclicity
Origin
Hal imported