I'm looking at the eigenfunction expansion of Brownian motion on the interval [0,1]:
$W_t = \sqrt{2} \sum_{k=1}^\infty Z_k \frac{\sin((k - \frac{1}{2}) \pi t)}{(k - \frac{1}{2}) \pi}.$
One deduces this fact by computing the eigenfunctions of the integral operator
$f(s) \to \int_0^1 \min(s,t)f(t)dt.$
Mercer's theorem says the eigenfunctions of this operator should form an orthonormal basis of $L^2[0,1]$. However, eigenfunctions of the integral operator are $\sqrt{2}\sin((k - \frac{1}{2}) \pi t)$, which clearly don't form a basis since they all satisfy $f(0) = 0$. What am I missing here?