  We consider estimation of a step function $f$ from noisy observations of a
deconvolution $\phi*f$, where $\phi$ is some bounded $L_1$-function. We use a
penalized least squares estimator to reconstruct the signal $f$ from the
observations, with penalty equal to the number of jumps of the reconstruction.
Asymptotically, it is possible to correctly estimate the number of jumps with
probability one. Given that the number of jumps is correctly estimated, we show
that the corresponding parameter estimates of the jump locations and jump
heights are $n^{-1/2}$ consistent and converge to a joint normal distribution
with covariance structure depending on $\phi$, and that this rate is minimax
for bounded continuous kernels $\phi$. As special case we obtain the asymptotic
distribution of the least squares estimator in multiphase regression and
generalisations thereof. In contrast to the results obtained for bounded
$\phi$, we show that for kernels with a singularity of order $O(|
x|^{-\alpha}),1/2<\alpha<1$, a jump location can be estimated at a rate of
$n^{-1/(3-2\alpha)}$, which is again the minimax rate. We find that these rate
do not depend on the spectral information of the operator rather on its
localization properties in the time domain. Finally, it turns out that adaptive
sampling does not improve the rate of convergence, in strict contrast to the
case of direct regression.
