A WSS random process $X(\mu,t)$ has $\eta_X(t) = 0$ and $R_X(\tau) = \sum_{k=1}^K\cos(\frac\pi k\tau)$ for a given positive integer $K$ in the interval $|t|<1$. Find the Karhunen-Loève expansion of $X(\mu,t)$ in the interval $(-1,1)$.
By definition $$ \lambda\phi(t_1)=\int_{-1}^1\sum_{k=1}^K\cos(\frac\pi k(t_1-t_2))\phi(t_2)\mathrm dt_2 $$ But I don't think the equation above can be solved, because after differentiating it twice $$ \lambda\phi''(t_1)=-\int_{-1}^1\sum_{k=1}^K\frac{\pi^2}{k^2}\cos(\frac\pi k(t_1-t_2))\phi(t_2)\mathrm dt_2 $$ I am thinking about directly expanding $R_X(t_1,t_2)$ and see whether or not I can find any trace of the eigenfunctions in it, but doesn't seem to be getting anywhere.
Any suggestions or hints are welcome.
I see what you mean. The general rule is that you cannot find the KL eigenfunctions in closed form, just like you cannot find eigenvectors (for $D>5$) in the discrete case. This is possible in a few special cases only. So you have to make an approximation. Two suggestions:
1) Do it numerically. No matter how ugly is your kernel the transformation is linear so you can represent it as a matrix and look for the eigenvectors.
2) Have you heared of Sirovich's snapshot method? Here you approximate the eigenfunctions as add-mixture of the process itself, i.e.
$$ \phi(t)=\int a(t-s)X(\mu,s)ds $$
and then you look for the coefficients $a$ which is much simpler than the original equation.
Hope this helps