I am currently diving into the theory of reproducing kernel Hilbert spaces and am just at the beginning of understanding the background of reproducing kernels. I have stumbled upon the following theorem firstly published by N. Aronszajn in Theory of reproducing kernels, 1950. It states:
If $K_i(x,y)$ is the reproducing kernel of class $F_i$ with norm $\Vert . \Vert_i$ for $i=1,2$ respectively, then $K(x,y)=K_1(x,y)+K_2(x,y)$ is the reproducing kernel of the class $F=\lbrace f=f_1+f_2 \vert f_1 \in F_1, f_2 \in F_2 \rbrace $ with norm $$ \Vert f \Vert = \min \lbrace \Vert f_1 \Vert_1 + \Vert f_2 \Vert_2 \mid f_1 \in F_1, f_2 \in F_2 \rbrace.$$
I completely understand the given proof of this statement but I currently still have problems with the intuition behind it. For my understanding what we do to prove this statement is to use the Hilbert space $F_1 \times F_2$ with inner product
$$ \langle (f_1,f_2),(g_1,g_2) \rangle = \langle f_1,g_1 \rangle _1 + \langle f_2,g_2 \rangle _2$$ and then filter away the nullspace of the addition operator $(f,g)\mapsto f+g$ for the case that $F_1 \cap F_2 \neq \lbrace 0 \rbrace$. At this point I am wondering why the direct sum (not necessarily the orthogonal direct sum) $$ F_1 + F_2 = \lbrace f_1 + f_2 | f_1 \in F_1, f_2 \in F_2 \rbrace$$ with the above defined scalar product (and induced norm $\Vert . \Vert = \Vert . \Vert_1 + \Vert . \Vert_2$) doesn't yield the RHKS for the kernel $K=K_1 +K_2$.
I suspect that something with the reproducing properties has to go wrong in this case but I cannot figure out what exactly it is, since we have
\begin{align}
i)\quad &K(.,y)=K_1(.,y)+K_2(.,y) \in F_1 + F_2 \quad \forall y \\
ii) \quad & \langle f_1+f_2,K(.,y) \rangle = \langle f_1+f_2,K_1(.,y)+K_2(.,y) \rangle \\
&= \langle f_1,K_1(.,y) \rangle _1+ \langle f_2,K_2(.,y) \rangle _2 = f_1(y)+f_2(y) \quad \forall y.
\end{align}
Thank you in advance!
Best Answer
If $F_1\neq F_2$ (as a normed space) and $F_1 \cap F_2 \neq \lbrace 0 \rbrace$ (as sets), you can choose functions $f=f_1+f_2\in F_1$, $g=g_1+f_2\in F_2$, with $f_2\in F_1 \cap F_2$.
The expression $\Vert h \Vert = \Vert h_1 \Vert_1 + \Vert h_2 \Vert_2$ (on $F_1+F_2$) is not well defined to $h=f+g=h_1+h_2$, because $h_1$ and $h_2$ can be writen as $h_1=f_1+2f_2$ and $h_2=g_1$ or $h_1=f_1$ and $h_2=g_1+2f_2$, for instance.
The Claim 1 in Norms and convergence on direct product of Banach spaces can helps.
Perhaps you find useful results searching for "\(\|h\|= \|h_1 \|_1 + \| h _2\|_2\)" on SearchOnMath.