Wiener's 'closure of translates' problem and Piatetski-Shapiro's uniqueness phenomenon

Abstract

Wiener characterized the cyclic vectors (with respect to translations) in $l^p(Z)$ and $L^p(R)$, $p=1,2$, in terms of the zero set of the Fourier transform. He conjectured that a similar characterization should be true for $1<p<2$. Our main result contradicts this conjecture.

Cited by in corpus (6)

• Characterization of informational completeness for covariant phase space observables
• Discrete Translates in Function Spaces
• $L^p$-integrability, dimensions of supports of fourier transforms and applications
• Translation-based completeness on compact intervals
• Cyclicity in ${\ell }^p$ spaces and zero sets of the Fourier transforms
• Completeness of Discrete Translates in $H^1(\mathbb{R})$