bounding norm of RKHS

Question

Suppose we have some RKHS H with respect to some continuous bounded Kernel K on the domain $X=[0,1]$ for example. Further suppose that $\sup_{x\in[0,1]}|f(x)|\leq B$. Is it possible to show $$ \|f\|_H \lesssim B \,\,\,?? $$

Answer 1

No, the statement is false in general. As a (counter)example take the Kernel $$ K(x,y) = \exp(-\lvert x - y\rvert)$$ and the functions $$ f_n(x)=\sin(2 \pi n x).$$ All those functions and the Kernel are continuous and bounded by 1 on $[0, 1].$ The RKHS associated with $K$ is Sobolev space on $[0,1]$ so $f_n\in\mathscr{H}_K$ for all $n$. This space has the norm $$ \Vert f \Vert^2_{\mathscr{H}_K}=\int_0^1 f(t)^2\,dt + \int_0^1 (f'(t))^2\,dt + f(0)^2+f(1)^2.$$ (see Theorem 1.3 of Saburou Saitoh, Yoshihiro Sawano).

Applying this to the $f_n$ of the example shows that $\Vert f_n \Vert_{\mathscr{H}_K}\rightarrow\infty$ as $n\rightarrow\infty.$