Let $\mathcal{H} (k), \mathcal{H} (K)$ be two RKHS of functions on the same set, X. It is clear that the intersection of these two spaces, with the norm $\| \cdot \| ^2 := \| \cdot \| _k ^2 + \| \cdot \| _K ^2 $ is also a RKHS.
Recently, I ran across a result in an online thesis which characterizes the positive kernel of this intersection space, Theorem 2.2.3, page 19 of: http://www.thesis.bilkent.edu.tr/0002953.pdf
The theorem states that if $G$ is the reproducing kernel of the intersection space, then, it is given as a quadratic form by:
$$ \sum \overline{\epsilon _x} \epsilon _y G(x,y) $$ is the infimum of $$ \sum \overline{\delta _x} \delta _y K(x,y) + \sum \overline{\alpha _x} \alpha _y k(x,y), $$ where $ \delta _x + \alpha _x = \epsilon _x, $ and where $\epsilon, \delta, \alpha$ are finitely supported functions in $X$.
However, I am having trouble following the proof, beginning at the top of page 21. I'd like to understand this, am I missing something?
Some more detail:
This intersection RKHS can be embedded isometrically into the direct sum of $\mathcal{H} (K)$ and $\mathcal{H} (k)$ under the map $f \mapsto f \oplus f$, call this isometry $W$. If $G$ denotes the reproducing kernel of the intersection space, then one can calculate $$ G_x = \frac{1}{2} W^* \left( K_x \oplus k_x \right). $$ Here $G_x$ is the point evaluation vector at $x \in X$ in $\mathcal{H} (G)$, i.e. if $f \in \mathcal{H} (G)$ then $\langle G_x , f \rangle = f(x)$ (inner product conjugate linear in first argument).
At the top of page 21 in the linked reference, the author calculates the norm of $\sum \epsilon _x G_x$ (finite sum, $\epsilon _x$ are complex coefficients) and this is the norm of $$h := W W^* g := W W^* \frac{1}{2} \sum \epsilon _x \left( K_x \oplus k_x \right). $$ Note that $WW^* $ is the orthogonal projection onto the range of the isometry $W$. At this point the author says this is the same as the norm of $$ \frac{1}{2} \sum \epsilon _x \left( K_x \oplus k_x \right) \oplus \frac{1}{2} \sum \epsilon _x \left( K_x \oplus k_x \right), $$ which I don't understand. There is at least some typo, as by the end of the calculation he concludes that the norm of $h = W W^* g$ is greater than that of $g$.
