I am faced with the problem of calculating the eigenfunctions for an operator of the form:
$(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $
Does anyone know for which functions (or types of functions) K analytic solutions for the eigenfunctions or eigenvalues are known? Or are there methods for solving this in particular cases?
After user1952009's comment I managed to make the following progress: using a Fourier transform on the equation the following form of a Fourier space eigenvalue equation can be obtained:
$ \hat{K}(\omega) \hat{f}(\alpha \omega) = \lambda \hat{f}(\omega) $
where $\hat{K}(\omega)$ is some function and $\lambda$ is the eigenvalue.
Does anyone know how to solve this equivalent equation?