Entire functions of exponential type and uniqueness conditions on their real part (Q1430443)

A classical uniqueness theorem due to Carlson asserts that if \(f\) is an entire function on the complex plane \({\mathbb C}\) of exponential type \(< \pi\) and \(f(n) = 0\) for all \(n \in\) \({\mathbb N} := \{\)the positive integers\(\}\) then \(f\) vanishes identically. R. P.~Boas proved in 1972 a related result for the real parts of entire functions which can be interpreted as a uniqueness result for harmonic functions: If \(f\) is entire and of exponential type \(< \pi\) and its real part vanishes on \({\mathbb Z} := \{\)the integers\(\}\) and \({\mathbb Z}+i\), then \(f\) is an imaginary constant. Later, in 1991, Trembinska studied the case of two complex variables, showing that if \(f\) is an entire function in \({\mathbb C}^2\) satisfying \(| f(z_1,z_2)| \leq Ce^{\tau (| z_1| +| z_2| )}\) (for all \((z_1,z_2) \in{\mathbb C}^2\) and some constants \(C > 0, \, 0 < \tau < \pi\)) such that its real part vanishes on \({\mathbb Z}^2\) and \(({\mathbb Z}+i)^2\), then \(f\) is identically zero, provided that: (i) the restriction of \(f\) to \({\mathbb R}^2\) belongs to \(L^2({\mathbb R}^2)\) and (ii) \(\sum | f(n_1,n_2)| < +\infty\), where the sum is extended to \({\mathbb Z}^2\). In this paper, the author proves that in the \(N\)-dimensional case the conclusion of the last statement still holds without (i) and that (ii) may be weakened. For this, she uses analytical functionals together with the Fourier-Borel transform defined mainly in the subspace of analytical functionals carried by a fixed compact subset of \({\mathbb C}^N\). Other important tools are difference and differential operators of infinite order with constant coefficients. In addition, Carlson-type results where the numbers \(f(n_1,n_2)\) satisfy some infinite order recurrence relations or assume prescribed interpolation values are also stated, including applications to some difference equations studied by Buck, Boas and Yoshino.