Consider a non-uniform ("generalized"?) convolution operator:
$$ A_h[f](t) = \int f(x)h(x,t)dx $$
I would like determine the eigenfunctions. In the "stationary" case where $h(x,t) = h(x-t)$ we have the standard convolution, whose eigenfunctions are the complex exponentials $e^{ikt}$ and eigenvalues are the Fourier transform of $h$.
Obviously this is a pretty open-ended problem. I'm specifically interested in the case that the convolution is just a spatially-varying Gaussian blur, so:
$$ h(x,t) = \exp\left[-\frac{(x-t)^2}{2\sigma_x^2}\right] $$
That is, the width of the blur depends on location. How can we compute the eigenfunctions, or equivalently diagonalize the non-uniform convolution operator matrix? I would guess that we could use wavelets (since waves work in the stationary case) but I suspect we need to impose limits on how quickly $\sigma_x$ changes (i.e. making it "locally stationary").
I would just add that a partial solution, or a link to a related solution, or even just an outline, would be really helpful!
It's very atypical to be able to compute the eigenfunctions of a general integral operator explicitly. What is special about convolution operators is that they exhibit translation symmetry, which is why periodic functions "diagonalize" them. Wavelets are known to almost diagonalize certain kinds of nonstationary operators - this is discussed in Meyer's book on wavelets and Calderon-Zygmund theory.
If your operator doesn't have some really nice symmetries (translational, rotational, etc), then you're pretty much stuck with making general existence claims, which fall under the heading of spectral theory. If you want to try and estimate the eigenvalues/eigenfunctions, you'll usually need to resort to numerical techniques.