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?



Sampling set: relatively dense and uniformly discrete
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](https://link.springer.com/article/10.1007/s13324-012-0027-4) ("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](https://books.google.co.il/books?hl=en&lr=&id=Y2CBBcDlo5kC&oi=fnd&pg=PP1&dq=introduction%20to%20nonharmonic&ots=EtbH0qrACP&sig=B7sTPUlL_JDUWOsuVP5qhaBG2JY&redir_esc=y#v=onepage&q=introduction%20to%20nonharmonic&f=false) "An Introduction to Non-Harmonic Fourier Series" in the case $n=1$. In the general case, one can extend a [result by Ingham](http://vhttps://link.springer.com/article/10.1007%2FBF01180426) (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](https://link.springer.com/content/pdf/10.1007/BF02395039.pdf) (appears in "Necessary density conditions for sampling and interpolation of certain entire functions"), and applies to both cases.


