  We study spectral subspaces of the Sturm-Liouville operator $f \mapsto
-(pf')'$ on $\mathbb{R}$, where $p$ is a positive, piecewise constant function.
Functions in these subspaces can be thought of as having a local bandwidth
determined by $1/\sqrt{p}$. Using the spectral theory of Sturm-Liouville
operators, we make the reproducing kernel of these spectral subspaces more
explicit and compute it completely in certain cases. As a contribution to
sampling theory, we then prove necessary density conditions for sampling and
interpolation in these subspaces and determine the critical density that
separates sets of stable sampling from sets of interpolation.
