  The well-known Asplund theorem states that the inverse of a (possibly
one-sided) band matrix $A$ is a Green matrix. In accordance with quasiseparable
theory, such a matrix admits a quasiseparable representation in its
rank-structured part. Based on this idea, we derive algorithms that compute a
quasiseparable representation of $A^{-1}$ with linear complexity. Many
inversion algorithms for band matrices exist in the literature. However,
algorithms based on a computation of the rank structure performed theoretically
via the Asplund theorem appear for the first time in this paper. Numerical
experiments confirm complexity estimates and offer insight into stability
properties.
