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I am reading this paper and met with this problem today. I am not familiar with reproducing kernel hilbert space, but after reading a lot, in my opinion it seems that here there are two groups basis:
$$
k(·,x_t)_{t=1}^{T}\;\text{ and }\;k_T(·,x_t)_{t=1}^{T}
$$
for the same hilbert functional space which consists of the linear combination of the basis. So what should we do is to find out the specific relation between the coefficients of two different basis for the same element. I just figured it out moments ago, and I happened to see your question, though eight years ago. Really time test, right?
I am tired so I just put my drafts here to answer it.
