Discretization theorems for entire functions of exponential type

Abstract

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