Discretization Theorems for Entire Functions of Exponential Type

TL;DR

This work establishes Marcinkiewicz–Zygmund type discretization inequalities for entire functions of exponential type (EFETs) on ${\mathbb R}^m$ and on cubes, linking $L_q$ norms to sampled values over carefully chosen sampling sets. The authors develop a rigorous framework using $\delta$-covering nets and $({\delta}_1,N)$-packing nets, prove sharp right- and left-hand bounds under precise geometric conditions, and extend discretization results to exponential polynomials via multivariate polynomial approximation and Ehrenpreis–Martineau representations. A key novelty is the removal of the assumption $f\in L_q({\mathbb R}^m)$ for the left inequality, achieved through a constructive approximation strategy with EFETs $f_n$ that converge on compacts and preserve exponential-type control. The results unify and broaden classical trigonometric and polynomial discretization theory, offering new tools for stable sampling and analysis of EFETs in high dimensions with explicit constants and growth conditions.

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_{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 $Ω$ 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.

Theorems & Definitions (50)

• Definition 1.1
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• Theorem 1.10