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I'm currently working on a problem in which I'm attempting to characterize a centered Gaussian random process \xi(x) on a manifold M given a known covariance function C(x,x') for that process. My current approach is to find a series expansion $\xi(x) = \sum_{n=1}^{\infty} X_n \phi_n(x)$ where the X_n's are Gaussian random variables with the standard distribution and the \phi_n(x)'s are the orthonormal basis functions of the reproducing kernel Hilbert space H generated by C(x,x'). I'm fairly certain that this procedure works so long as C^{-1}(x,x') exists, since that allows me to define the inner product for the RKHS.

However, I'm not sure how to proceed regarding finding an orthonormal basis for the RKHS. My RKHS should be some subset of L^2(M), so I assume that if I have an orthonormal basis on L^2(M) I should be able to find one for H. Is there a general procedure for this?

Thanks,

Aimless

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# Finding an orthonormal basis for a reproducing kernel Hilbert space.

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