  We establish adaptive results for trend filtering: least squares estimation
with a penalty on the total variation of $(k-1)^{\rm th}$ order differences.
Our approach is based on combining a general oracle inequality for the
$\ell_1$-penalized least squares estimator with "interpolating vectors" to
upper-bound the "effective sparsity". This allows one to show that the
$\ell_1$-penalty on the $k^{\text{th}}$ order differences leads to an estimator
that can adapt to the number of jumps in the $(k-1)^{\text{th}}$ order
differences of the underlying signal or an approximation thereof. We show the
result for $k \in \{1,2,3,4\}$ and indicate how it could be derived for general
$k\in \mathbb{N}$.
