Entire Functions of Exponential Type and Uniqueness Conditions on Their Real Part

Abstract

This paper extends uniqueness results due to Boas and Trembinska, on entire functions with exponential growth whose real part vanishes on lattice points. Here the case is studied where the real part of the function satisfies given relations or assumes prescribed values at lattice points. These results are obtained thanks to such tools as analytic functionals, their Fourier-Borel transform and some operators acting in the space of entire functions in C N of exponential type, including difference and differential operators of infinite order with constant coefficients. There are also applications to some difference equation studied by Buck, Boas and Yoshino.