Frame-type families of translates

Abstract

We construct a uniformly discrete, and even sparse, sequence of real numbers $\Lambda=\{\lambda_n\}$ and a function g in $L^2(R)$, such that for every q>2, every function f in $L^2(R)$ can be approximated with arbitrary small error by a linear combination $\sum c_n g(t-\lambda_n)$ with an $l_q$ estimate of the coefficients: \|\{c_n\}\|_{l_q}\leq C(q)\|f\|. This can not be done for q=2, according to [2].