In detail, $\mathcal{H}$ is a reproducing kernel hilbert space (RKHS), and $f^*$ is fixed.
Problem 1 is $$f(x) \in \underset{g\in \mathcal{H}}{arg min} \Vert f^*(x) -g(x)\Vert_{\mathcal{H}}$$
Problem 2 is
$$f'(x) \in \underset{g\in \mathcal{H}}{arg min} \Vert f^*(x) -g(x)\Vert_{\mathcal{L}_2}$$
Are $f(x)$ and $f'(x)$ the same?
In my understanding, it should be the same but I only have a vague feeling. Since $$\Vert f^*(x) -g(x)\Vert_{\mathcal{L}_2}\leq\Vert f^*(x) -g(x)\Vert_{\mathcal{H}}K(x,x)$$ and $$\Vert f^*(x) -g(x)\Vert_{\mathcal{H}}\leq\Vert f^*(x) -g(x)\Vert_{\mathcal{L}_2}\Vert P_{K,X}\Vert_{\mathcal{L}_2}$$ where $P_{K,X}$ is the power function from the book $\textit{Scattered Data Approximation}$.