On~the uniqueness of an expansion in generalized eigenfunctions of a~differential operator

V. A. Tkachenko

Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 455-466

Résumé

The space $\mathscr E_\rho$ of the entire functions of order $\rho$ ($1\rho\infty$) with the usual topology and the operator $\mathscr L$, induced by a differential operation $l[y]=y^n+p_{n-2}(z)y^{n-2}+\dots+p_0(z)y$, $n>1$, and “boundary” conditions $F_i[y]=0$ ($i=1,\dots,n$), where the $F_i$ are linear functionals on $\mathscr E_\rho$. Conditions are indicated under which the formal expansion $f\sim-\Sigma_\lambda\operatorname{Res}(\mathscr L-\lambda E)^{-1}f$ uniquely determines an element $f\in\mathscr E_\rho$. As a corollary it is established that if $\Delta(\lambda)=\Sigma c_k\lambda^k\in\mathscr E_\mu$, $\mu>1$, has an infinite number of zeros and $f(z)\in\mathscr E_\rho$, $\rho\mu(\mu-1)$, then $f(z)\equiv0$ whenever $$ \sum^\infty_{k=1}\frac{c_k(\lambda^{k-1}f(0)+\dots+f^{(k-1)}(0))}{\Delta(\lambda)} $$ is an entire function. Bibliography: 10 titles.