Zero sets of entire absolutely monotonic functions (Q2782547)

The main object studied in the paper is the class of entire functions \(f:{\mathbb C}\rightarrow {\mathbb C}\) which admit the following integral representation NEWLINE\[NEWLINE f(z) = \int_0^{+\infty} e^{zu} P(d u), \quad z \in {\mathbb C}, NEWLINE\]NEWLINE where \(P\) is a non-negative finite Borel measure on \({\mathbb R}^+ = [0, +\infty)\). Such functions are called absolutely monotonic. The main result is the following theorem. NEWLINENEWLINENEWLINETheorem. Let \(E = \{b_k\}_{k=1}^\infty \subset \{ z: \operatorname {Re}z\geq 0\}\) be a set of complex numbers with no accumulation points. Then \(E\) is the zero set of an entire absolutely monotonic function if and only if it satisfies the following conditions: (a) it is symmetric with respect to the real axis and no element of \(E\) is located on this axis; (b) for every \(\omega \geq 0\), the following is true NEWLINE\[NEWLINE \sum_{b_k \in E_\omega}{|\operatorname {Re}b_k|+ 1 \over |b_k|^2 +1} < \infty, \quad \quad E_\omega = \{b\in E : \operatorname {Re}b \leq \omega\}. NEWLINE\]NEWLINE This is an extension of a similar statement proved by \textit{I. V. Ostrovskij} [Theory Probab. Appl. 33, No. 1, 167-171 (1988); translation from Teor. Veroyatn. Primen. 33, No. 1, 180-184 (1988; Zbl 0672.60029)].NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].