Uniqueness theorems for entire functions of exponential type (Q798797)

Let f be an entire function of exponential type \(\tau <\pi\), \(| f(m)| <M,\) and \(Re f(m)=0\) for \(m=0,\pm 1,\pm 2,... .\) Then if \(\sum| Im f(m)| <\infty\) and if either \(Re f(m+i)\quad or\quad Im f(m+i)=0,\) except on a set of density less than 1-(\(\tau\) /\(\pi)\), then \(f(z)\equiv 0.\) The results do not hold if \(\tau =\pi\). The proofs are by an application of Fourier analysis.