On the growth of the entire functions of exponential type with given zeros along a line
The main result of this paper is the following theorem: Let \(\Lambda=\{\lambda_ n\}\) be a sequence of complex numbers such that \(|\text{Re}(\lambda_ n)|\geq\delta|\lambda_ n|\), \(\delta>0\), and let \(g\) be an entire function of exponential type whose zeros satisfy the same condition. There exists an entire nonzero function \(f\) of exponential type such that \(f_{\mid\Lambda}=0\) and \(| f(iy)|\leq| g(iy)|\), \(y\in\mathbb{R}\), iff there exists a constant \(M\) such that for all numbers \(r\) and \(R\), \(0<r<R<\infty\), we have: \[ \max\{\sum_{{r\leq|\lambda_ n|<R}\atop{\text{Re} \lambda_ n<0}}- \text{Re}(1/\lambda_ n),\sum_{{r\leq|\lambda_ n|<R}\atop{\text{Re} \lambda_ n>0}}\text{Re}(1/\lambda_ n)\}\leq{1\over 2\pi} \int_ r^ R {\log| g(iy)g(-iy)| \over y^ 2}dy+M. \] This result sums up several previous results of Carlson, Carleman, Fuchs, Leont'ev, Levin, Malliavin and Rubel, making additional assumptions on \(\Lambda\) or/and the zero set of \(g\).
scientific article; zbMATH DE number 61278
scientific article