On small perturbations of the set of zeros of functions of sine type

B. Ya. Levin ; I. V. Ostrovskii

Izvestiya. Mathematics , Tome 14 (1980) no. 1, pp. 79-101

Résumé

A function of sine type means an entire function $S(z)$ of exponential type $\sigma>\nobreak0$, satisfying the condition $0$ outside some strip $|\operatorname{Im}z|\nobreak H$. With the normalization $S(0)=1$ these functions can be represented in the form \begin{equation} S(z)=\lim_{R\to\infty}\prod_{|\lambda_k|}(1-z\lambda_k^{-1}). \end{equation} Let $\widetilde S(z)$ denote the function obtained from $S(z)$ by replacing $\lambda_k$ by $\lambda_k+\psi_k$ in (1), where $\{\psi_k\}$ is a bounded sequence. In this paper necessary and sufficient conditions on $\{\psi_k\}$ are found, under which $\widetilde S(z)$ is also a function of sine type. Expressions for $\widetilde S(z)$ in terms of $S(z)$ are obtained in the case where $\psi_k=a_1\lambda_k^{-1}+\dots+a_n\lambda_k^{-n}+b_k\lambda_k^{-n}$, where $\{b_k\}\in L^p$, $p>1$. Bibliography: 9 titles.