On the approximation order of principal shift-invariant subspaces of \(L_ p(\mathbb{R}^d)\) (Q1379648)

For \(1\leq p\leq\infty\), sufficient conditions on the generators \(\{\varphi_h\}_{h> 0}\) are given which ensure that the \(h\)-dilates of the shift-invariant space generated by \(\varphi_h\) provide \(L_p\)-approximation of order \(k>0\). Examples where \(\varphi_h\) is an exponential box spline or certain dilates of the Gaussian \(e^{-|\cdot |^2}\) are considered; it is shown that our sufficient condition then provides an optimal lower bound on their approximation order.