A uniqueness theorem for entire functions

Abstract

Let $G(k)={\int }_0^{1}g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem. {\bf Theorem}. {\it If ${\mathrm{limsup}}_{k\to +\infty }|G(k)|<\infty$, then $g=0$. There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that $G(k_j)=0$, $k_j<k_{j+1}$, ${\lim }_{j\to \infty }{k}_j=\infty$, ${\lim }_{k\to \infty }|G(k)|$ does not exist, ${\mathrm{limsup}}_{k\to +\infty }|G(k)|=\infty$. This $g$ oscillates infinitely often in any interval $[1-\delta ,1]$, however small $\delta >0$ is.}