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

Abstract

We construct a uniformly discrete sequence $\{\hat{I}{»}_1<\hat{I}{»}_2<\cdots \}\subset \mathbb{R}$ 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 \[ f(x) = \sum_{n=1}^{\infty} \langle f, g_n^* \rangle \, g(x-λ_n) \] convergent in the $L^2(\mathbb{R})$ norm.

To appear in the Journal of Functional Analysis