  We propose a new notion of variable bandwidth that is based on the spectral
subspaces of an elliptic operator $A_pf = - (pf')'$ where $p>0$ is a strictly
positive function. Denote by $c_{\Lambda} (A_p)$ the orthogonal projection of
$A_p$ corresponding to the spectrum of $A_p$ in $\Lambda$, the range of this
projection is the space of functions of variable bandwidth with spectral set in
$\Lambda $.
  We will develop the basic theory of these function spaces. First, we derive
(nonuniform) sampling theorems, second, we prove necessary density conditions
in the style of Landau. Roughly, for a spectrum $\Lambda = [0,\Omega] $ the
main results say that, in a neighborhood of $x\in R$, a function of variable
bandwidth behaves like a bandlimited function with local bandwidth $(\Omega /
p(x))^{1/2}$.
  Although the formulation of the results is deceptively similar to the
corresponding results for classical bandlimited functions, the methods of proof
are much more involved. On the one hand, we use the oscillation method from
sampling theory and frame theoretic methods, on the other hand, we need the
precise spectral theory of Sturm-Liouville operators and the scattering theory
of one-dimensional Schr\"odinger operators.
