Heads up Im a physicist!
I want to know explicitly the eigenfunctions of the 1D gaussian kernel$$ K(x,y) = e^{-(x-y)^2/\sigma^2} $$when it is integrated, that is$$ (Kf)(n,x)=\int_{-\infty}^{\infty}e^{-(x-y)^2/\sigma^2}f(n,y)dy = \lambda(n) f(n,x), $$where $x,y\in \mathbb{R}$ and $n$ is a possibly continuous label of the $n$th eigenfunction and eigenvalue. I’m open
What is the explicit form of $f$ and $\lambda$? I have been reading that they are related to harmonic functions in a hypersphere (?) but I didn’t understand it.
Perhaps relevant links
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