Near the beginning of the last century Hermann Weyl examined the problem of expressing a general function on a half-line as a combination of eigenfunctions of a Sturm-Liouville operator with asymptotically constant coefficients. Weyl's solution was later re-examined and generalized by Kodaira, who applied modern operator-theoretic techniques to the problem. I will describe the work of Weyl and Kodaira, together with its relation to problems in representation theory, where Harish-Chandra used Weyl's theorem as a guide to his own work on the Plancherel formula. Finally I will describe a new, more geometric, approach, which has the advantage of fitting very well with representation-theoretic applications. This is joint work with Qijun Tan.