The fractional Brownian motion is a centered Gaussian process with the following covariance function (covariogram):
$E[B(t)B(s)]=C(\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H})$
Consider the 2D-fBm, in which case $\Vert\cdot\Vert$ is the 2D-norm and $t,s\in\mathbb{R}^2$. I would like to get the Karhunen-Loeve transform of the 2D-fBm, and for this I need to know the eigenfunctions or eigenvectors of the covariance matrix of the fBm. This amounts to solving this equation:
$\int (\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H} )\phi(t)dt=\lambda \phi(s) $, where $H\in(0,1)$.
I couldn't find this result and I will appreciate any assistance in finding their analytical formula.
Thanks, Ido.