I'm currently trying to determine the spectral representation of covariance kernels with the structure: $$K(s,t) = C \cdot [F(\min\{s,t\}) - F(s)F(t)], \quad s,t \in \mathbb{R}$$ of centered and square-integrable stochastic processes, with $C$ being a constant and $F$ a continuous function to satisfy the assumptions of Mercer's theorem. I already determined the series representation for the covariance kernel of a Brownian bridge on $[0,1]$ with $K_B(s,t) = \min\{s,t\}-st = \sum_{j=1}^\infty \lambda_j f_j(s)f_j(t)$, $\lambda_j = (j\pi)^{-2}$ and $f_j(t) = \sqrt{2}\sin(j\pi t)$.
It follows that $K(s,t) = C\cdot K_B(F(s),F(t))$ if $F$ is increasing (e.g. a continuous distribution function). Does that also imply that the structure of eigenfunctions is perhaps similar to $f_j(F(t))$? If so, is there a proper argument to validate this conjecture?
The classic approach with solving the integral equation $$\lambda f(s) = \int_\mathbb{R} K(s,t)f(t)dt$$ doesn't lead to any conclusion since I don't assume that $F$ admits a density and even if, the solution cannot be computed similarly to the Brownian bridge or motion. I believe my only chance here would be to replace $F$ with an estimator e.g. the empirical distribution function $F_n$ in the case of $F$ being a d.f. Is there any other way to solve or attack this problem?
This is a similar question to this one that unfortunately wasn't answered: Eigen function of one Stochastic Process from the eigen function of another Stochastic Process