Sampling set: relatively dense and uniformly discrete

Question

The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$

We say that a discrete set $\Lambda\subset\mathbb{R}^d$ is sampling for $PW_\Omega$ if there exists a constant $C>0$ such that for all $f\in PW_\Omega$ $$\|f\|_{L^2(\mathbb{R}^d)}\leqslant C\|f\|_{\ell^2(\Lambda)}$$ where $\|f\|_{\ell^2(\Lambda)}:=\left(\sum_{\lambda\in\Lambda}|f(\lambda)|^2\right)^{1/2}$.

We say that $\Lambda$ is uniformly discrete if the separation $$\delta(\Lambda):=\inf_{\lambda,\lambda'\in\Lambda,\lambda\neq\lambda'}|\lambda-\lambda'|$$ is positive. And we say that $\Lambda$ is relatively dense if the gap $$\rho(\Lambda):=\sup_{x\in\mathbb{R}^d}\inf_{\lambda\in\Lambda}|x-\lambda|$$ is finite.

I have read that a sampling set is always relatively dense, and that if further it is uniformly discrete then the converse inequality also holds i.e. there exist a constant $C'>0$ such that $$\|f\|_{\ell^2(\Lambda)}\leqslant C'\|f\|_{L^2(\mathbb{R}^d)}$$ holds for all $f\in PW_\Omega$. Does anyone knows a reference or a quick proof of this two facts?

Answer 1

When asking this question, in principle one should make a distinction between two cases:

1. When $n=1$, the space $PW_{S}$ is a space of entire functions of exponential type, with all of their unique properties.

2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results were obtained by Olevskii and Ulanovskii ("On multi-dimensional sampling and interpolation").

   A proof to the fact that a uniformly discrete set forms a Bessel sequence can be found in thm 17, on chapter 2 in Young's book "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a result by Ingham (thm 2, "Some trigonometrical inequalities with applications to the theory of series") together with the fact that the inequality $$\sum_{\lambda\in\Lambda}|f(\lambda)|^2 \leq C||f||^2_{L^2(\mathbb R^n)}$$ holds for all $f\in PW_S$ is equivalent to having the inequality $$||\sum_{\lambda\in\Lambda}c_\lambda e^{i\lambda t}||^2_{L^2(S)}\leq C\sum_{\lambda\in\Lambda}|c_\lambda|^2$$ hold for all finite sequence of coefficients $\{c_\lambda\}$.
   The fact that a sampling set is always relatively dense follows from Landau's necessary condition (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.