Quasi-sampling sets for analytic functions in a cone (Q1279879)

The authors study the analogues of sampling sets for analytic functions in cones of \(\mathbb C^n\). Cartwright-type and Bernstein-type theorems, previously known only for functions of exponential type in \(\mathbb C^n\), are extended to the case of functions of arbitrary order in a cone. Let \(E\) be a dense set of order \(\rho\geqslant 1\) relative to the positive hyperoctant \(\mathbb R_+^n.\) Then for every \(\eta\in (0, 1)\) each function \(f\in H(W_\rho; \rho, \infty),\) which is bounded in a neighborhood of the origin and bounded on \(E\) is bounded on the cone \(C(\eta).\) Moreover, for each \(\sigma\in (0, \infty)\) and \(f\in H(W_\rho; \rho, \sigma)\) there exists a positive constant \(\varDelta = \varDelta(E, \rho, \sigma, \eta),\) such that \[ \sup_{x \in C(\eta)\setminus B(0, R)} |f(x)|\leqslant \varDelta \sup_{x\in E}|f(x)| \] for some \(R = R(f,\eta) < \infty.\) The corresponding result for \(\varepsilon\)-nets is also given. The uniqueness result only holds if the cone \(W_\tau,\) in which a function is defined, is wide enough. Let a set \(E\) be an \(\varepsilon()\)-net of order \(\rho>0\) for some cone \(C(\eta_0)\) and let \(\lim_{|x|\to\infty}\sup_{x\in E}|x|^{-\rho}\log|f(x)|=-\infty\) for some function \(f\in H(W_\tau;\rho,\infty)\), \(\tau\in (1, \rho).\) Then \(f\equiv 0.\)