The typical form of Karhunen-Loeve expansion is on a detrended stochastic process. E.g., let $Y(t)$ be a stochastic process on $[0,T]$, and let $X(t) = Y(t)-\mathbb{E}Y(t)$ with a continuous covariance function $R_X(s,t)$, then there exists a series of orthonormal functions $\phi_k(t)$ on $[0,T]$ and independent zero-mean and normalized random variables $Z_k$ such that
$$ X(t) = \sum_{k=1}^{\infty} Z_k \sqrt{\lambda_k} \phi_k(t) $$
where $\lambda_k$ and $\phi_k(t)$ are the solutions of the following eigensystems equation:
$$ \int_0^T R_X(s,t)\phi_k(t)dt=\lambda_k\phi_k(s) $$
My question is, if we do not subtract the mean from the process, and just focus on the raw autocorrelation function $R_Y(s,t)$, as $R_Y(s,t)$ is also positive definite symmetric functions in $s,t$, it has a spectral decomposition (Mercer's theorem)
$$ R_Y(s,t) = \sum_k \mu_k \psi_k(s)\psi_k(t) $$
Can we expand the original process $Y(t)$ in $\psi(t)$? And does this decomposition enjoy some similar properties of the original K-L decomposition, such as the optimal compaction of energies of the signal in the eigenfunctions?
$$ Y(t)=\sum_k \tilde{Z}_k \sqrt{\mu_k} \psi_k(t) $$
The empirical reason is that sometimes we are interested in the information contained in the mean as well, not only in the fluctuation around the mean. While the original K-L expansion optimally compacts the total variance into the first few eigenfunctions $\phi_k(t)$, the direct decomposition of $Y(t)$ may optimally compact the total fluctuation around $0$, i.e., the raw energy of the signal: $\mathbb{E}\int_0^T |Y(t)|^2dt$ into $\psi_k(t)$. Has it been shown to be true? And if true, is the result trivial in the sense that $\psi_k(t)$ and $\phi_k(t)$ are connected in some simple way?
It is trivial to see that \begin{align} R_X(t,s)&=E[(X(t) - E[X(t)])(X(s)-E[X(s)])] \\ &= E[(Y(t) - E[Y(t)])(Y(s)-E[Y(s)])] = R_Y(t,s). \end{align} Since both Mercer's theorem and the KL decomposition use an eigendecomposition of $R_X$ (and thus $R_Y$), we have that $\psi_k$ and $\phi_k$ are actually the same vectors. In order to find a decomposition of the non-centered process $Y(t)$ it suffices to notice that $\phi_k$ is a complete orthonormal basis of some Lebesgue space on $[0,T]$ (e.g. $L^2$) - in most cases - and therefore denoting $y(t) = E[Y(t)]$ there exist scalars $\{y_k\}$ such that $$ y(t) = \sum_{k} y_k \sqrt{\lambda_k} \phi_k(t). $$ You can then write the process $Y(t)$ as $$ Y(t) = X(t) + y(t) =\sum_k (Z_k+y_k)\sqrt{\lambda_k}\phi_k. $$