extended-DMD (e-DMD) additionally approximates eigenfunctions of the Koopman operator[@WilliamsKernel; @WilliamsEDMD.2015].

### Sparsity.

The concept of sparsity is useful to find solutions of underdetermined systems of linear equations. A solution $x \in \mathbb{C}^n$ is considered *sparse*, when most of the entries of $x$ vanish,[@Cands2014MathematicsOS].

Next to offering numerical advantages, a solution with a limited number of terms facilitates the manual interpretation of the results. The problem of finding sparse solutions can be connected to model reduction techniques, as discussed in [@loiseau_brunton_2018].

[@Donoho.2006] showed that an underdetermined system of equations can be converted to a convex problem using a sparsity promoting condition. Consider solving $$y = \theta s^{\prime}$$ for $s^{\prime}$, with $y \in \mathbb{R}^n$, $\theta \in \mathbb{R}^{n \times m}$, $s^{\prime}   \in \mathbb{R}^m$ and $m>n$. The parameter vector $ s^{\prime}$ needs to be determined in order to fit the measurement vector $y$ using the product of the observation matrix $\theta$ and $ s^{\prime}$.

The common $\text{L}_2$-regularized regression promotes a high number of involved terms. Contrariwise, by using the $\text{L}_1$-norm as regularization term, terms of minor impact are neglected from the solution. The $\text{L}_1$-regularized regression (see Eq. ) with weight $\lambda$ can be solved e.g. by the LASSO described by [@TibshiraniLasso]. The solution $s$ 