Discretization Theorems for Entire Functions of Exponential Type

Abstract

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 $Ω=\left\{X_ν\right\}_{ν=1}^\iy\subset\R^m$ for the right inequality to be valid and a sufficient condition on $\hat{I}\text{c◯}$ for the left one to hold true. In addition, $L_\iy(Q^m_b)$-discretization inequalities on an $m$-dimensional cube are proved for entire functions of exponential type and exponential polynomials.

39 pages