# Kernel function from polynomial basis functions

In chapter 3 & 6 of Bishop's Pattern Recognition and Machine Learning, he showed that the equivalent kernel based on eqn (3.62)

$$k(x,x') = \beta \phi(x)^T (\alpha I + \beta \Phi^T \Phi )^{-1}\phi(x')$$

and the standard kernel based on the feature space mapping eqn (6.10),

$$k(x,x') = \phi(x)^T \phi(x') = \sum_{j=1}^M \phi_j(x)\phi_j(x')$$

both of which have a sinc like behavior corresponding to Figures 3.11(left) and 6.1(left), see below.

This is quite impossible given that when $x=0$, all $\phi_j(x)=x^j=0$, which implies $\phi(x)^T\phi(x')=0\ \forall x'$. Is my assumption that $\phi_j(x)=x^j$ wrong?

My understanding is that polynomial basis are global functions, so how can one arrive at the conclusion that its kernel have a local behavior, it's not obvious to me at this point.

• This question will not be answerable unless you reproduce the necessary context, such as equations and figures. – Sycorax Jul 5 '18 at 0:10
• Added equations and figures. – David KWH Jul 5 '18 at 15:41
• Thank you! The question is now in the "reopen" queue, where reviewers will voted on whether the question is sufficiently clear to be re-opened. (I have voted to re-open the question.) – Sycorax Jul 5 '18 at 16:06