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Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a real-valued symmetric, positive definite function. Let $\{(x_i,y_i)\}_{1\leq i\leq N}\subset[0,1]^2$ be a finite, distinct set of coordinates. The point-wise evaluation matrix $A\in\mathbb{R}^{N\times N}$ is defined by,

$ A_{jk} = f(x_j,x_k). $

What can be said about the distribution of the eigenvalues of the matrix $A$ assuming (i) $f$ is analytic or (ii) $f\in H^k([0,1]^2)$ for some $k\geq1$?

Note that any change of order of the coordinates produces a similar matrix to $A$ and hence the same eigenvalues.

Thanks in advance.

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  • $\begingroup$ What exactly do you want to know? It is a little hard to tell. $\endgroup$
    – Igor Rivin
    Commented Sep 12, 2011 at 14:34
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    $\begingroup$ You should say more about how the points $\{x_j\}_{j=1}^{N}$ are chosen. Are you interested in some probabilistic answers? Are they chosen independently of each other? If yes, you should look into the answers given by random matrix theory. At least for the global statistics their methods might be adaptable. $\endgroup$
    – Helge
    Commented Sep 12, 2011 at 16:10
  • $\begingroup$ Have you already looked at the case of the very well-studied posdef function $f(x,y)=e^{-\gamma(x-y)^2}$ to gain some insight? $\endgroup$
    – Suvrit
    Commented Sep 13, 2011 at 6:54

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