  Adaptive spectral (AS) decompositions associated with a piecewise constant
function $u$ yield small subspaces where the characteristic functions
comprising $u$ are well approximated. When combined with Newton-like
optimization methods for the solution of inverse medium problems, AS
decompositions have proved remarkably efficient in providing at each nonlinear
iteration a low-dimensional search space. Here, we derive $L^2$-error estimates
for the AS decomposition of $u$, truncated after $K$ terms, when $u$ is
piecewise constant and consists of $K$ characteristic functions over Lipschitz
domains and a background. Our estimates apply both to the continuous and the
discrete Galerkin finite element setting. Numerical examples illustrate the
accuracy of the AS decomposition for media that either do, or do not, satisfy
the assumptions of the theory.
