Yury Belov

In 1966 D. Newman and H. Shapiro posed the following problem. Let $G$ be a function from Fock space $\mathcal{F}$ and such that $e^{zw}G\in\mathcal{F}$ for any $w\in\mathbb{C}$. Is it true that $$Span{FG; FG ∈ F} = Span{e^{wz}G : w ∈ C}?$$ Recently the author (joint with A.Borichev) has constructed a counterexample to this conjecture. On the other hand, we are able to show that if $G$ satisfies some regularity conditions, then conjecture holds. These results are clоsеly connected to some spectral synthesis problems in Fock space.