Do we have Karhunen–Loève expansion for White Noise? 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?
 A: The process you describe (independent normal r.v.'s at every point) is not white noise. So there are really two questions here.


*

*Does the process you describe have a KL expansion? The answer is no. One (informal) reason is that this process requires uncountably many Gaussian random variables to be described, while by definition, a KL expansion only uses countably many of them. Another (related) reason is that this process cannot be realised as a Borel measure on any reasonable function space. (It can be realised as a measure on the space of all functions endowed with its product $\sigma$-algebra, but this is not a Borel measure on the space of all functions endowed with the product topology.)

*Does white noise have a KL expansion? The answer is yes, provided that you are willing to work in spaces of distributions rather than functions. White noise $\xi$ is a random distribution with $\mathbf{E} \xi(s)\xi(t) = \delta(t-s)$ and can be realised as $\xi = \sum_{n=1}^\infty e_n\, \xi_n$ for a sequence of i.i.d. Gaussians $\xi_n$ and any orthonormal basis $e_n$ of $L^2([0,1])$. This sum doesn't converge pointwise of course, but it converges for example in every Sobolev space $H^s$ for $s < -1/2$.
A: Not in the case of white noise.  If you managed to represent $W_t = \sum e_i(t)Z_i$ then the $W_t$ must live on a countable generated sigma field, on which e.g. $L^2$ is seperable, whereas the collection $W_t$ exhibit an uncountable orthonormal set.  However, the continuity is not particularly important.  If you have expanded the kernel as $\sum \phi_i(t)\phi_i(s)$ it assures that $\sum \phi_i^2(t) < \infty $ so that $\sum Z_i \phi_i(s)  < \infty $, but as long as you can do that you are in good shape.  For example if $K(s,t) = 1, s, t < \frac 12$ or $s,t > \frac 12$ and 0 otherwise the process can be written $e_1(t) Z_1 + e_2(t)Z_2$ where the $Z_i$ are i.i.d. normal and $e_1(t) = 1_{(0,\frac 12)}(t)$, $e_2(t) = 1_{ (\frac 12,1)}(t)$
