A uniqueness theorem for entire functions of two complex variables (Q1176304)

The main result of the paper is the following Theorem 1. Let \(f(z_ 1,z_ 2)\) be an entire function of exponential type less than \(\pi\), and let \(f(x_ 1,x_ 2)\) belong to \(L^ 2\). Let \(\text{Re} f(m,n)=0\) for all integers \(m\), \(n\) and \(\sum_ n\sum_ m|\text{Im} f(m,n)|<\infty\). If \(\text{Re}f(m+i,n+i)=0\) for all integers \(m\), \(n\), then \(f(z_ 1,z_ 2)\equiv 0\).