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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?

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 facts?

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?