Exercise 12.7: Different Kernels for Polynomial Functions
(a)
For $k_1$, use the Binomial theorem: \begin{equation} k_1(x, y) = \sum_{l=0}^m \frac{m!}{l!(m - l)!}x^l y^l. \end{equation} That $k_2$ is positive semidefinite is obvious. That they generate polynomial functions of degree at most $m$ follows from that all linear combinations of the kernels at polynomial functions of degree at most $m$, and the subspace of polynomials functions of degree at most $m$ contains all pointwise limits: convergence at $m+1$ points implies convergence of the coefficients, which implies that the limit is also a polynomial of degree at most $m$.
(b)
The norms that the kernels generate are different.