On the expansion of an entire function of finite order in the eigenfunctions of a~differential operator

V. A. Tkachenko

Sbornik. Mathematics, Tome 18 (1972) no. 4, pp. 559-570

Résumé

In this paper we consider the operator $L$ which is induced in the space $\mathscr E_p$ of entire functions of order $\rho$ by the operator $l[y]=y^{(n)}+p_{n-2}y^{(n-2)}+\dots+p_0y$ and the boundary conditions $F_i[y]=0$, $i=1,2,\dots,n$. Here $p_{n-2}(z),\dots,p_0(z)$ are polynomials and $F_i(y)$ is a linear functional in $\mathscr E_p$. We establish the completeness of the eigenfunctions of the operator $L$, show the possibility of expansion in terms of these eigenfunctions, and estimate the rate of convergence of such an expansion. Bibliography: 10 titles.