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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.

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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...)

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  • $\begingroup$ Thank you for your answer, it clarifies. But may I request few comments regarding the difference between "pick interpolation" and "interpolation space", specifically how what is the difference of their application areas. $\endgroup$
    – Creator
    Aug 20, 2015 at 18:55
  • $\begingroup$ Interpolating functions, as in "Pick interpolation" or "Lagrange interpolation", is simply to find a function $\phi$ that takes prescribed values at some given points and satisfies some criterion such as being holomorphic with a bound on $|\phi'|$ (Pick-Nevanlinna), or being a polynomial of a given degree (Lagrange), or minimizing some quadratic functional (splines). This last one makes use of RKHS, as well as Pick, but "interpolation spaces" are defined as a continuous family of Hilbert or Banach spaces that "interpolate" between two given spaces, an almost unrelated concept. $\endgroup$ Aug 21, 2015 at 13:21

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