Discretization Theorems for Entire Functions of Exponential Type

We prove $L_q({\mathbb{R}}^m)$–discretization inequalities for entire functions $f$ of exponential type in the form \ba C_2|f|{L_q(\R^m)} \le \left(\sum{ν=1}^\iy \left\vert f\left(X_ν\right) \right\vert^q\right)^{1/q} \le C_1|f|_{L_q(\R^m)},\qquad q\in[1,\iy], \ea with estimates for $C_1$ and $C_2$. We find a necessary and sufficient condition on $\mathrm{\Omega }={\left\{X_{\nu }\right\}}_{\nu =1}^{\text{iy}}\subset {\mathbb{R}}^m$ for the right inequality to be valid and a sufficient condition on $\mathrm{\Omega }$ for the left one to hold true. In addition, $L_{\text{iy}}(Q_b^m)$-discretization inequalities on an $m$-dimensional cube are proved for entire functions of exponential type and exponential polynomials.