# Eigenfunctions and eigenvalues of the exponential kernel

What are the eigenfunctions and the eigenvalues of the exponential kernel?

The exponential kernel is defined as $$k(x,x')=\sigma^2\exp\left(\frac{||x-x'||}{l}\right)$$ where both $\sigma>0$ and $l>0$.

Mercers theorem tell us that for every kernel function $k(x,x')$ there exists a decomposition in eigenfunctions $\phi_i(x)$ and corresponding eigenvalues $\lambda_i$ such that

$$k(x,x')=\sum_{i=1}^\infty \lambda_i \phi_i(x)\phi_i(x)$$

The Fourier transform $$\mathcal{F}(k)(\omega)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} k(r) e^{i\omega r}dr$$ of the function $$k(r)=\sigma^2 \exp\left(\frac{||r||}{l}\right)$$ with $r=x-x'$ is $$\mathcal{F}(k)(\omega)=\frac{\sqrt{\frac{2}{\pi }} \sigma^2 l}{l^2 \omega ^2+1}.$$ How to proceed from here?

• Some information may be found in Gaussian Processes for Machine Learning, section 4.3, "Eigenfunction Analysis of Kernels." – Sycorax says Reinstate Monica Jan 13 '16 at 19:15
• Is this a machine learning question or a pure math question? – gung - Reinstate Monica Jan 13 '16 at 20:28
• @gung It could be both. I was however hoping that the solution to this problem is better known in the machine learning problem. That's why I posted it here. – Julian Karls Jan 14 '16 at 12:14