Frame-type families of translates

TL;DR

The paper constructs a sparse sequence of real numbers and a function in L^2(R) such that functions in L^2(R) can be approximated with small error using linear combinations with controlled l_q norms of coefficients for q>2, but not for q=2.

Contribution

It introduces a specific sparse sequence and a function enabling approximation in L^2(R) with l_q coefficient control for q>2, highlighting a limitation at q=2.

Findings

Approximation possible for q>2 with controlled l_q coefficients.

Construction of a sparse sequence and a function in L^2(R).

Impossibility of such approximation for q=2.

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].