The setup is the following, suppose you have a set of data $(y_{t_1},x_{t_1}),...,(y_{t_n},x_{t_n})$. You believe that $x_t$ is predictive of $y_t$ such that
$$y_t = f(x_t)+\epsilon_t$$
for some function $f$, where $\epsilon_t \sim N(0,\sigma^2)$, and $t \in T (= [a,b]$, WLOG).
Define a Mercer kernel $K$ and consider a corresponding Gaussian random function on $L^2(T)$.
Now, let $f^k$ be the projection of $f$ onto the space $V_k$ spanned by the eigenfunctions $\phi_1,\phi_2,\ldots,\phi_k$.
Is $f^k$ a finite-dimensional normal RV? If so, what is its distribution (in particular, its variance)?
Moreover, show that $\mathbb{E}[f^k \mid \{x_{t_1},\ldots,x_{t_n},y_{t_1},\ldots,y_{t_n}\}]$ is the solution to
$$ \min_{\psi \in V_k} \sigma^{-2} \sum_i \mid\psi(x_{t_i})-y_{t_i}\mid^2 + \sum_{j=1}^k \lambda_k^{-1} (\langle\psi,\phi_j\rangle)^2 $$
The last bit looks to be ridge regression, but I'm not sure how to link it all together. Any ideas?