This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space. As I understand Mercer theorem confirms the existence of RKHS with appropriate kernel. My understanding is that the kernel takes two vector and returns a scalar. Therefore RKHS is nice for classification type application. My question is can we extend the idea to interpolation? In other words if the 'kernel' can take a matrix and a vector it would return a function or a vector. Can we have a similar RKHS space for this case? Is this a valid question? In case it is not, would you please let me know what is the problem in this view?
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1$\begingroup$ Most likely all of your answers are in Alger and McCarthy's "Pick Interpolation and Hilbert function spaces book". If you had a specific problem understanding kernels and RKHSs you should ask it on mathstackexchange. $\endgroup$– Chris RamseyCommented Aug 16, 2015 at 17:35
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$\begingroup$ @ChrisRamsey Do you think OP question is related to "Interpolation space' or "interpolation of operators"? $\endgroup$– CreatorCommented Aug 16, 2015 at 21:34
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$\begingroup$ @Creator I'm not entirely clear on what is being asked as far as interpolation in the original question. There is definitely a notion of vector-valued RKHS and I was just wanting to point towards a good book. $\endgroup$– Chris RamseyCommented Aug 16, 2015 at 22:38
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$\begingroup$ @ChrisRamsey Thank you for clarification. In any case do you think is there a relation between vector-valued RKHS and "interpolation spaces"? $\endgroup$– CreatorCommented Aug 17, 2015 at 0:09
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$\begingroup$ @Creator I'm not sure. $\endgroup$– Chris RamseyCommented Aug 17, 2015 at 18:11
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