On Cartwright's theorem (Q2817001)

A characterization of the sets for which Cartwright's theorem holds true is presented. Let \(E_{<\sigma}\), resp. \(E_{\sigma}\), be the class of all entire functions of exponential type less than \(\sigma\), resp. less than or equal to \(\sigma\). A set \(\Lambda\subset\mathbb{R}\) is called uniformly discrete (u.d.) if \(d(\Lambda):=\inf\limits_{\lambda,\lambda'\in\Lambda,\lambda\neq\lambda'} |\lambda-\lambda'|>0\). Denote by \(D^-(\Lambda)\) the lower uniform density of \(\Lambda\). A set \(\Lambda\subset\mathbb{R}\) is called a Cartwright set (CS) for a class \(M\) of entire functions, if there is no function \(f\in M\) which is bounded on \(\Lambda\) and unbounded on the real axis. A theorem of M. L. Cartwright states that the set of integers is a CS for the class \(E_{<\pi}\). The authors prove that a set \(\Lambda\subset\mathbb{R}\) is a CS for \(E_{<\sigma}\) if and only if it contains a u.d. subset \(\Lambda^*\) satisfying \(D^-(\Lambda^*)\geq\sigma/\pi\). The connection between these sets and sampling sets of entire functions of exponential type is also discussed. Let \(B_{\sigma}\) denote the subclass of \(E_{\sigma}\) of functions \(f\) bounded on the real axis. A set \(\Lambda\) is called a sampling set (SS) for \(B_{\sigma}\) if there is a constant \(K\) such that \(\|f\|_\infty\leq K\|f|_\Lambda\|_\infty\), for all \(f\in B_{\sigma}\). It is proved that a set \(\Lambda\subset\mathbb{R}\) is an SS for \(E_{\sigma}\) if and only if it is an SS for \(B_{\sigma}\). It is shown that for the classes \(E_{\sigma}\) the opposite is also true: a set \(\Lambda\subset\mathbb{R}\) is an CS for \(E_{\sigma}\) if and only if it is an SS for \(E_{\sigma}\).