Spectral Subspaces of Sturm-Liouville Operators and Variable Bandwidth

We study spectral subspaces of the Sturm-Liouville operator $f\mapsto -(p{f}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$ 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 $1\mathrm{/}\sqrt{p}$. 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.