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

We construct a uniformly discrete sequence $\{{\lambda }_1<{\lambda }_2<\cdots \}\subset \mathbb{R}$ and functions $g$ and $\{g_n^{\ast }\}$ in $L^2(\mathbb{R})$, such that every $f\in L^2(\mathbb{R})$ admits a series expansion $$f(x)=\sum _{n=1}^{\mathrm{\infty }}⟨f,g_n^{\ast }⟩g(x-{\lambda }_n)$$ convergent in the $L^2(\mathbb{R})$ norm.