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

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