Is there a generalization of Sturm-Liouville theory to higher order ODEs?

Question

In informal terms, the Sturm-Liouville theorem states that if you have a second-order ODE with certain homogeneous boundary conditions on the interval [a,b] in Sturm-Liouville form, and if you manage to find the set of eigenfunctions, g(x), that solve both the ODE and boundary conditions, then if you have an arbitrary function f(x) on that same interval you should be able to write f(x) as a linear combination of all your g(x)s. The constants can be found using the abbreviated formula <f,g>/<g,g>. My questions is this: is there any generalization of Sturm-Liouville theory to 3rd order linear ODEs or 4th or nth order? Do the same principles for second-order ODEs that I discussed above have some sort of analogue in higher order ODEs? Because it seems like any time I see discussion of Sturm-Liouville theory, it is always invoked in the context of second-order ODEs.