Approximation from locally finite-dimensional shift-invariant spaces

After exploring some topological properties of locally finite-dimensional shift-invariant subspaces $S$ of $L_p(\mathbb {R}^s)$, we show that if $S$ provides approximation order $k$, then it provides the corresponding simultaneous approximation order. In the case $S$ is generated by a compactly supported function in $L_\infty (\mathbb {R})$, it is proved that $S$ provides approximation order $k$ in the $L_p(\mathbb {R})$-norm with $p>1$ if and only if the generator is a derivative of a compactly supported function that satisfies the Strang-Fix conditions.