Schauder frames of discrete translates in $L^2(\mathbb{R})$

Abstract: We construct a uniformly discrete sequence ${\lambda_1 < \lambda_2 < \cdots} \subset \mathbb{R}$ and functions $g$ and ${g_n^*}$ in $L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ admits a series expansion [ f(x) = \sum_{n=1}^{\infty} \langle f, g_n^* \rangle \, g(x-\lambda_n) ] convergent in the $L^2(\mathbb{R})$ norm.