# Reproducing kernels: how do I numerically compute the decomposition?

Suppose I'm given a kernel,

$$K(x,y) : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$$

In order to describe/understand the (unique) associated RKHS, I seek its eigenfunctions, as per Mercer's theorem, which should give a "feel" for the space. How do I do this?

• For generality, I want a numerical procedure.
• Presumably, this involves discretising a part of $$\mathbb{R}$$ and using eig or eigh from Numpy or Matlab.
• But I guess I need to know the inner product first. How? (Numerically, again, is ok).
• What if, as for the Gaussian (RBF) kernel, $$K(x,y) = e^{-\frac{\|x - y\|^2}{2\sigma^2}}$$, the inner product is defined in Fourier space (see p28)