Suppose a random function $X(t)$ can be written with the help of the Karhunen-Loève expansion (or some other basis expansion) by $$X(t) = \sum_{k=1}^\infty \xi_k \psi_k(t),$$ where $\psi_1(t), \psi_2(t),\dots$ are orthonormal eigenfunctions corresponding to the non-negative eigenvalues $\lambda_1,\lambda_2,\dots$ of the covariance operator $\Gamma$ and the random variables $\xi_k = \int_{a}^bX(s)\psi_k(s)ds$ are the uncorrelated scores with $E(\xi_k)=0$, $E(\xi_k \xi_l)=0$ for $k\neq l$ and $E(\xi_k^2)=\lambda_k$.
I am interested in the derivative $X'(t)$ of $X(t)$. Suppose the eigenfunctions $\psi_k(t)$ are differentiable, then, a natural approach would be to use the basis expansion above to get
$$X'(t) = \sum_{k=1}^\infty \xi_k \psi_k'(t).$$
This approach doesn't work always. For example, in the case where $X(t)$ is a Wiener process, $\psi_k(t)=\sqrt{2}\sin((k-1/2)\pi t)$ and $\lambda_k \sim 1/k^2$. However, while the derivative of $\psi_k$ exists and is obviously bounded, it is well known that the Wiener process is a.s. not differentiable.
Why can't I see this fact from the derivative of the expansion? Which property am I missing?
(What irritates me additionally is the fact that the variance of the right hand side, $E((\sum_{k=1}^\infty \xi_k \psi_k'(t))^2) = \sum_{k=1}^\infty \lambda_k \psi_k'(t)^2 \leq const \sum_{k=1}^\infty 1/k^2$, is bounded?!)
Can someone enlighten me why this natural approach doesn't work in general and/or give me some conditions when it works?
This is very analogous to Fourier series expansion. Try expanding a straight square wave in a Fourier series, and you will see the Gibbs phenomenon. It essentially comes down to the fact that you'll have pointwise convergence, but not uniform convergence. The Dirichlet conditions give you some limit on the sorts of behavior allowed in your $X(t)$ in order to get better behavior in the convergence. That, at least, works for Fourier series; however, any sort of eigenvector expansion such as you're employing, especially if it's an orthonormal expansion, will behave much like a Fourier series.