Let $\mathcal{H}\subset \mathbb{R}^X$ a real reproducing kernel Hilbert space (RKHS) with reproducing kernel $k : X\times X\rightarrow\mathbb{R}$.
Is then $\mathcal{H}$ closed under absolute value, that is do we have that
$$\tag{1} f\in \mathcal{H} \quad\text{implies}\quad |f|\in\mathcal{H}\ ?$$
(Here, $|f| : X \rightarrow\mathbb{R}, \ x\mapsto |f(x)|$, is simply $f$ composed with the absolute value $|\cdot|$ on $\mathbb{R}$.)
If $(1)$ does not hold generically, are there conditions on $k$ for which it is true?