Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: <https://en.wikipedia.org/wiki/Interpolation_space>. My question are these related somehow? Can we define both under the same framework? I have seen some papers link given below but could not clarify my doubt. 


<http://www.ams.org/journals/mcom/2008-77-263/S0025-5718-07-02096-0/S0025-5718-07-02096-0.pdf>


<http://arxiv.org/pdf/1402.0099.pdf>


Any clarification would be appreciated.



Is there relation between vector valued RKHS and interpolation space?
Vector-valued or scalar-valued doesn't make a big difference, as well as real or complex. If $\mathcal{H}\_1\subset\mathcal{H}\_0$ are RKHSs, $\mathcal{H}\_1$ dense in
$\mathcal{H}\_0$, and $(e\_n)\_{n\in\mathbb{N}}$ is an orthonormal basis of $\mathcal{H}\_0$, the reproducing kernel of $\mathcal{H}\_0$ is $h\_0(x,y)=\sum e\_n(x)\bar{e}\_n(y)$ (tensor product in the vector-valued case). And if the $e\_n$s are also orthogonal in $\mathcal{H}\_1$ (so that $e\_n/||e\_n||\_{\mathcal{H}\_1}$ is an orthonormal basis of $\mathcal{H}\_1$, whose reproducing kernel is then $h\_1(x,y)=\sum e\_n(x)\bar{e}\_n(y)/||e\_n||\_{\mathcal{H}\_1}^2$), then the interpolation space $\mathcal{H}\_\theta$ ($0\le\theta\le1$) is a RKHS with orthonormal basis $e\_n/||e\_n||\_{\mathcal{H}\_1}^\theta$ .


("Interpolation" as in "Pick interpolation" is not the same as in "interpolation space", which might be confusing at first...)


