О спектральном синтезе в одном топологическом векторном пространстве целых функций

S. S. Platonov

Problemy analiza, no. 3 (1996), pp. 97-117.

Résumé

Let $\mathcal{F}$ be the set of all entire functions $f(z), z=x+iy$, such that $\mathrm{sup}_{|y|\le l}|f(x+iy)|,(1+x^{2})^{-k}\infty$ for all $l>0$. $\mathcal{F}$ is a locally convex space with respect to certain topology. It is proved that every closed invariant under derivation linear subspace $\mathcal{H}\subseteq \mathcal{F}$ is the closed span of the functions $z^{k}e^{i\lambda z}, \lambda\in C, k\in N\cup \{\infty \}$. A set $\wedge=\{\lambda \in C:e^{i\lambda z}\in \mathcal{H}\}$ is called the spectrum of $\mathcal{H}$, and we suppose that $\lambda$ contains in $\wedge$ with the multiplicity $r(\lambda)$, where $r(\lambda)=\mathrm{inf}\{k\in N\cup \{\infty \}:z^{k}e^{i\lambda z}\notin \mathcal{H}$. The complete description of various spectrums of such subspaces $\mathcal{H}$ is obtained.