Let $W$ be a random process (my White Noise) on $[-1,1]$ such that:
- $W(t)$ is a normal random variable with mean $0$ and standard deviation $1$ for all $t \in [-1,1]$
- $E(W(t)W(s)) = 0$ for all $t, s \in [-1,1]$ with $t \neq s$.
The covariance function is $$ K(t,s) = E(W(t)W(s)) = \begin{cases} 1 & \text{if } t = s \\ 0 & \text{otherwise} \end{cases} $$
Karhunen–Loève Theorem states that if the covariance function $K(t,s)$ is continuous there exists a Karhunen–Loève expansion. However, the above covariance function is NOT continuous.
Is there any "Karhunen–Loève"-like expansion in this case? More generally, do we have some kinds of orthogonal expansion if the covariance function is not continuous?