Let $\Omega$ be an open subset of $\mathbb{R}^d$. Under regularity conditions, we know that the $s$-th order Sobolev space $H^s(\Omega)$ with $s \geq d/2$ is a reproducing kernel Hilbert space. In contrast, $H^s(\Omega)$ with $s < d/2$ is only a Hilbert space without the reproducing property.

My question is about the construction of orthonormal basis of $H^s(\Omega)$.

For $s \geq d/2$, the eigenfunction of the reproducing kernel gives us an orthonormal basis, which up to a rescaling of magnitudes is also an orthonormal basis of $L^2(\Omega)$ (which can be equivalently written as $H^0(\Omega)$).

For $s < d/2$, how do we construct an orthonormal basis in a similar fashion, given now there is no reproducing kernel? **Moreover, for $0 \leq s_1, s_2 < d/2$, is it possible to align the orthonormal bases of $H^{s_1}(\Omega)$ and $H^{s_2}(\Omega)$ so that they only differ up to a rescaling of magnitudes?**