We consider estimating an unknown signal, which is both blocky and
sparse, corrupted by additive noise. We study three interrelated
least squares procedures and their asymptotic properties. The first
procedure is the fused lasso, put forward by Friedman et al. (2007),
which we modify into a different estimator, called the fused
adaptive lasso, with better properties. The other two estimators we
discuss solve least squares problems on sieves, one constraining the
maximal 
norm and the maximal total variation seminorm, the
other restricting the number of blocks and of the number of nonzero
coordinates of the signal. We derive conditions for the recovery of
the true block partition and the true sparsity pattern by the fused
lasso and the fused adaptive lasso, and convergence rates for the
sieve estimators, explicitly in terms of the constraining
parameters.