Prediction bounds for (higher order) total variation regularized least squares

We establish oracle inequalities for the least squares estimator $\hat f$ with penalty on the total variation of $\hat f$ or on its higher order differences. Our main tool is an interpolating vector that leads to lower bounds for compatibility constants. This allows one to show that for any $N \in \mathbb{N}$ the $N^{\rm th}$ order differences penalty leads to an estimator $\hat f$ that can adapt to the number of jumps in the $(N-1)^{\rm th}$ order differences.