Weighted-$L^2$ interpolation on non-uniformly separated sequences

We define a weighted-$\ell^2$-norm associated to a discrete sequence $\Gamma$ in $\mathbb{C}$ and a weight function $\varphi$. We then give a sufficient condition which ensures that we can always extend weighted-$\ell^2$ data to global holomorphic functions which are also weighted-$L^2$. The condition is such that the so-called upper density of $\Gamma$ is strictly less than one.