Karhunen-Loève expansions

As seen in chaos expansions kind of, and used in low rank kriging, and assumed in PCA and implied by many basis decompositions etc.

Suppose we have a collection $\{{\phi }_n\}$ of real-valued functions on our index space $T$, and a collection $\{{\xi }_n\}$ of uncorrelated random variables. Now we define the random process $$f(t)=\sum _{n=1}^{\mathrm{\infty }}{\xi }_n{\phi }_n(t).$$ We might care about the first two moments of $f$, i.e. $$\mathbb{E}\{f(s)f(t)\}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_n^2{\phi }_n(s){\phi }_n(t)$$ and variance function $$\mathbb{E}\{f^2(t)\}=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_n^2{\phi }_n^2(t)$$

Now suppose that we have a stochastic process where the index $T$ is a compact domain in ${\mathbb{R}}^N$. The corresponding expansion of $f$ in the above form is known as the Karhunen-Loève expansion. Suppose that $f$ has covariance function $K$ and define an operator $\mathcal{K}$, taking the space of square integrable functions on $T$ to itself, by $$(\mathcal{K}\psi )(t)={\int }_{T}K(s,t)\psi (s)ds$$ Suppose that ${\lambda }_1\ge {\lambda }_2\ge \dots$, and ${\psi }_1,{\psi }_2,\dots$, are, respectively, the (ordered) eigenvalues and normalised eigenfunctions of the operator. That is, the ${\lambda }_n$ and ${\psi }_n$ solve the integral equation $${\int }_{T}K(s,t)\psi (s)ds=\lambda \psi (t)$$ with the normalisation $${\int }_T{\psi }_n(t){\psi }_m(t)dt=\{\begin{array}{ll}1& n=m\\ 0& n\ne m\end{array}$$ These eigenfunctions lead to a natural expansion of $K$, known as Mercer’s Theorem, which states that $$K(s,t)=\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n{\psi }_n(s){\psi }_n(t)$$ where the series converges absolutely and uniformly on $T\times T$. The Karhunen-Loève expansion of $f$ is obtained by setting ${\phi }_n=$ ${\lambda }_n^{\frac{1}{2}}{\psi }_n$, so that $$f_t=\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n^{\frac{1}{2}}{\xi }_n{\psi }_n(t).$$ Now, when does such an expansion exist?

Because I look at signals on unbounded domains a lot, the case where a countable basis cannot be assumed to exist seems important.