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 $\{\hat{I}{»}_n{\}}_{n=1}^{\infty }$ satisfying $|\hat{I}{»}_n|=n+o(1)$, 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 \[ f(x) = \sum_{n=1}^{\infty} g_n^*(f) g(x-λ_n) \] convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative.

To appear in Revista Matemática Iberoamericana