Going back through some old class notes, I found this statement that I haven't been able to prove or determine if it's false and only the result of bad note-taking.
Statement: Given arbitrary n-degree polynomials, if you Gram-Schmidt with respect to the Sturm-Liouville weight gives us orthogonal polynomials that are eigenfunctions of the Sturm-Liouville differential equation.
Attempt: I do recall that solutions to Sturm-Liouville problems form a complete orthogonal basis and that they are unique, up to a constant. But all this just gives me the the orthogonalized functions can be written as a linear combination of the eigenfunctions that span a subspace of space span by all eigenfunctions.
Any hints forward (or against the above statement) are welcome.