A uniqueness theorem for entire functions

TL;DR

This paper proves a uniqueness theorem for entire functions generated by integrals of $L^1$ functions, showing that boundedness of the integral transform implies the function is zero, and constructs non-zero examples with specific oscillatory properties.

Contribution

It establishes a new uniqueness criterion for entire functions related to integral transforms and constructs non-zero functions with prescribed oscillatory behavior.

Findings

If the integral transform is bounded, the function must be zero.

Existence of non-zero functions with unbounded integral transforms and infinite oscillations.

Characterization of oscillatory behavior near the endpoint 1.

Abstract

Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem. {\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, 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, $\limsup_{k\to +\infty}|G(k)|=\infty$. This $g$ oscillates infinitely often in any interval $[1-\delta, 1]$, however small $\delta>0$ is.}