Let $w(x,y)$ be a real-valued symmetric function over the $[0,1]$x$[0,1]$ interval. We also know that if $n$ is an integer, and you pick $n$ values $x_i$ in the $[0,1]$ interval, then the matrix $\Sigma_N$ whose entry $(i,j)$ is given by $w(x_i,x_j)$ is symmetric positive definite. To me, $\Sigma_N$ is a covariance matrix so I call $w$ a covariance function.
What is the determinant of $w$? How can it be defined, and more importantly calculated for a given function $w$?
This is as far as I've gotten: If the matrix is diagonal, that is if $w(x,y)=0$ if $x\ne y$, then
$\det \Sigma_N = \prod_{i=1}^N w(x_i,x_i) $ so that when $N$ tends to infinity and taking $x_i$ points with even spacing and $x_1=0$ and $x_N=1$ $\lim_{N\to +\infty} \frac{1}{N-1} \log\det\Sigma_N = \int_0^1 \log w(x,x)\text{d} x$ How can you generalize that to arbitrary covariance functions?
I'm no mathematician, feel free to make this more rigorous.