Quasi-greedy systems of integer translates (Q999276)

The authors study finitely generated shift-invariant subspaces of \(S\subset L_2(\mathbb{R}^d)\) of the form \[ S=\overline{\text{span}\{\varphi(\cdot-k) : \varphi\in \Phi, k\in \mathbb{Z}^d\}}, \] where \(\Phi\) is a finite seet. They show that if this space has a Schauder basis of the form \(\{\psi(\cdot-k) : \psi\in \Psi, k\in \mathbb{Z}^d\}\) that is quasi-greedy (which means that if one chooses the terms in the series expansion in an order so that the absolute values of the coefficients are not increasing, then one gets convergence), then this basis is in fact a Riesz basis.