# Gaussian kernel eigenfunctions

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references.

What is the eigenfunction of a multivariate Gaussian kernel: \begin{equation} f(x,y) = \exp\left(-\frac{\lVert x - y\rVert^2}{2\sigma^2}\right) \end{equation}

I am interested in the eigenfunctions with respect to $L^2$ norm: \begin{equation} \int f(x, y) v_i(y) dy = \lambda_i v_i(x) \end{equation} and also with respect to a Gaussian probability distribution $p(x)$: \begin{equation} \int f(x, y) v_i(y) p(y) dy = \lambda_i v_i(x) \end{equation} I am aware of the numerical approximation to the problem like Nystrom method. However, it should be possible to find a closed form solution when the probability distribution is also Gaussian.

• $\nu(x)=1$ --- are there more? – Carlo Beenakker Dec 23 '13 at 23:28

In the notation of the question, $$\int f(x,y)\nu(y)dy=W_\sigma\ast \nu(x)$$ where $W_\sigma(x)=e^{-|x|^2/2\sigma^2}$ and $\ast$ denotes convolution. Thus, if $$\int f(x,y)\nu(y)dy=\lambda\nu(x)$$ then by taking Fourier transforms: $$\lambda \hat{\nu}= \widehat{\lambda \nu} =\widehat{W_\sigma\ast \nu} =\widehat{W_\sigma}\widehat{\nu} =c_1W_{c_2}\hat{\nu}$$ since the Fourier transform of a Gaussian is a Gaussian which is rescaled in mass and variance from the original one by some constants $c_1,c_2>0$ whose exact values are easily computable but unimportant. So evidently $\hat{\nu}$ must be zero in the complement of a sphere centered at $0$, since the spheres centered at $0$ are the sets on which the multiplier equality $\lambda=c_1W_{c_2}$ can hold.
Now as for $L^2$ eigenfunctions, there are none other than zero... because any such eigenfunction would have to be the inverse Fourier transform of an $L^2$ function which is supported in a sphere and since spheres have measure zero, $0\in L^2$ is the only such function.
Now for the second question, the operator $$\nu\mapsto \int f(\cdot, y)\nu(y) p(y)dy$$ for a centered Gaussian $p$ is not translation invariant and so will not be given by a Fourier multiplier. I would guess that even in this case there can be no nonzero $L^2$ eigenfunctions but I can't think of a decent proof at the moment. However, I suspect that you may have intended to ask instead for eigenfunctions of the standard operator $$\nu\mapsto \int f(\cdot, y)\nu(y) dy$$ which are in $L^2(pdx)$ - in that case there are lots of them. Any harmonic function of polynomial growth and the inverse Fourier transform of $f d\mu$ where $f$ is smooth and $\mu$ is uniform measure on a sphere centered at zero, for instance.
The second question, regarding with respect to Gaussian distribution $$p(x)$$, has been answered in closed form at this link.