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

Abstract

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 $\{g_n^{\ast }{\}}_{n=1}^{\infty }$ on $L^p(\mathbb{R})$, such that every $f\in L^p(\mathbb{R})$ admits a series expansion

convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative.