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

For every $p>(1+\sqrt{5})\mathrm{/}2$ we construct a uniformly discrete real sequence $\{{\lambda }_n{\}}_{n=1}^{\mathrm{\infty }}$ satisfying $\mathrm{\mid }{\lambda }_n\mathrm{\mid }=n+o(1)$, a function $g\in L^p(\mathbb{R})$, and continuous linear functionals $\{g_n^{\ast }{\}}_{n=1}^{\mathrm{\infty }}$ on $L^p(\mathbb{R})$, such that every $f\in L^p(\mathbb{R})$ admits a series expansion $$f(x)=\sum _{n=1}^{\mathrm{\infty }}{g}_n^{\ast }(f)g(x-{\lambda }_n)$$ convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative.