Discretization Theorems for Entire Functions of Exponential Type

TL;DR

This paper establishes discretization inequalities for entire functions of exponential type in multiple dimensions, providing necessary and sufficient conditions on sampling sets for these inequalities to hold.

Contribution

It introduces new discretization inequalities for entire functions of exponential type and characterizes sampling sets for these inequalities in multiple dimensions.

Findings

Discretization inequalities hold with explicit constants.

Necessary and sufficient conditions for sampling sets are identified.

Results extend to exponential polynomials on cubes.

Abstract

We prove $L_q(\R^m)$--discretization inequalities for entire functions $f$ of exponential type in the form \ba C_2\|f\|_{L_q(\R^m)} \le \left(\sum_{\nu=1}^\iy \left\vert f\left(X_\nu\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 $\Omega=\left\{X_\nu\right\}_{\nu=1}^\iy\subset\R^m$ for the right inequality to be valid and a sufficient condition on $\Omega$ 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.